Probability · Mathematics · MHT CET

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MCQ (Single Correct Answer)

1

The probability that a non leap year selected at random will contain 52 Saturdays or 53 Sundays is

MHT CET 2025 26th April Evening Shift
2

A fair $n$ faced die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$, then $\mathrm{n}=($ where $\mathrm{n} \in \mathbb{N})$

MHT CET 2025 26th April Evening Shift
3

A fair coin is tossed a fixed number of times. If the probability of getting 5 tails is same as the probability of getting 7 tails, then the probability of getting 3 tails is

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4

The probability that a person is not a sportsperson is $\frac{1}{6}$. Then the probability that out of the 6 members of the family, 5 are sportspersons is

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5

The cumulative distribution function of a discrete random variable X is

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X}=x & -4 & -2 & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \mathrm{~F}(\mathrm{X}=x) & 0.1 & 0.3 & 0.5 & 0.65 & 0.75 & 0.85 & 0.90 & 1 \\ \hline \end{array} $$

then $\frac{P(X \leqslant 0)}{P(X>0)}=$

MHT CET 2025 26th April Morning Shift
6

The following is p.d.f. of continuous random variable X

$$ \mathrm{f}(x)= \begin{cases}\frac{x}{8} & , \text { if } 0 < x < 4 \\ 0 & , \text { otherwise }\end{cases} $$

Then $F(0.5), F(1.7)$ and $F(5)$ is respectively

MHT CET 2025 26th April Morning Shift
7

A box contains 8 red and $x$ number of green balls. 3 balls are drawn at random, if the probability that 3 balls being red is $\frac{7}{15}$, then number of green balls is…

MHT CET 2025 26th April Morning Shift
8

A doctor assumes that patient has one of three diseases $\mathrm{d} 1, \mathrm{~d} 2$ or d 3 . Before any test he assumes an equal probability for each disease. He carries out a test that will be positive with probability 0.7 if the patient has disease $\mathrm{d} 1,0.5$ if the patient has disease d 2 and 0.8 if the patient has disease d3. Given that the outcome of the test was positive then probability that patient has disease d2 is

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9

The probability that a student is not a swimmer is $\frac{1}{5}$. The probability that out of 5 students selected at random 4 are swimmers is

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10

A player tosses two coins. He wins ₹ 10 , if 2 heads appears, ₹ 5 , if one head appear and ₹ 2 if no head appears. Then variance of winning amount is

MHT CET 2025 25th April Evening Shift
11

Consider the probability distribution

$$ \begin{array}{|l|l|l|l|l|l|} \hline \mathrm{X}=x & 1 & 2 & 3 & 4 & 5 \\ \hline \mathrm{P}(\mathrm{X}=x) & \mathrm{K} & 2 \mathrm{~K} & \mathrm{~K}^2 & 2 \mathrm{~K} & 5 \mathrm{~K}^2 \\ \hline \end{array} $$

Then the value of $\mathrm{P}(\mathrm{X}>2)$ is

MHT CET 2025 25th April Evening Shift
12

Let X denote the number of hours you study on a Sunday. It is known that

$$ \mathrm{P}(\mathrm{X}=x)=\left\{\begin{array}{cc} 0.1 & , \text { if } x=0 \\ \mathrm{k} x & , \text { if } x=1 \text { or } 2 \\ \mathrm{k}(5-x) & , \text { if } x=3 \text { or } 4 \\ 0 & , \text { otherwise } \end{array}\right. $$

where k is constant. Then the probability that you study at least two hours on a Sunday is

MHT CET 2025 25th April Morning Shift
13

A pair of fair dice is thrown 4 times. If getting the same number on both dice is considered as a success, then the probability of two successes are

MHT CET 2025 25th April Morning Shift
14

A family has 3 children. The probability that all the three children are girls, given that at least one of them is a girl is

MHT CET 2025 25th April Morning Shift
15

If a random variable X has the following probability distribution of X

$$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \mathrm{X}=x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mathrm{P}(\mathrm{X}=x) & 0 & \mathrm{k} & 2 \mathrm{k} & 2 \mathrm{k} & 3 \mathrm{k} & \mathrm{k}^2 & 2 \mathrm{k}^2 & 7 \mathrm{k}^2+\mathrm{k} \\ \hline \end{array} $$

Then $P(x \geq 6)=$

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16

If $X \sim B\left(6, \frac{1}{2}\right)$, then $P(|X-2| \leqslant 1)=$

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17

A random variable X has following p.d.f. $\mathrm{f}(x)=\mathrm{kx}(1-x), 0 \leqslant x \leqslant 1 \quad$ and $\quad \mathrm{P}(x>\mathrm{a})=\frac{20}{27}$, then $\mathrm{a}=$

MHT CET 2025 23rd April Evening Shift
18

If A and B are independent events such that $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right)=\frac{3}{25}$ and $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=\frac{8}{25}$, then $P(A)=$

MHT CET 2025 23rd April Evening Shift
19

A fair coin is tossed 100 times. The chance of getting a head even number of times is

MHT CET 2025 23rd April Morning Shift
20

In a game a man wins $Rs \,\, 40$ if he gets 5 or 6 on a throw of a fair die and loses ₹ 20 for getting any other number on the die. If he decides to throw the die either till he gets a five or six or to a maximum of three throws, then his expected gain/loss (in rupees) is

MHT CET 2025 23rd April Morning Shift
21

Bag I contains 3 red and 2 green balls and Bag II contains 5 red and 3 green balls. A ball is drawn from one of the bag at random and it is found to be green. Then the probability that it is drawn from Bag I is

MHT CET 2025 23rd April Morning Shift
22

If a random variable $X$ has p.d.f. $f(x)=\left\{\begin{array}{ll}\frac{a x^2}{2}+b x & , \text { if } 1 \leqslant x \leqslant 3 \\ 0 & , \text { otherwise }\end{array}\right.$ and $f(2)=2$, then the values of $a$ and $b$ are, respectively

MHT CET 2025 22nd April Evening Shift
23
Three urns respectively contain 2 white and 3 black, 3 white and 2 black and 1 white and 4 black balls. If one ball is drawn from each um, then the probability that the selection contains 1 black and 2 white balls is
MHT CET 2025 22nd April Evening Shift
24

In a box containing 100 apples, 10 are defective. The probability that in a sample of 6 apples, 3 are defective is

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25

Four defective oranges are accidentally mixed with sixteen good ones. Three oranges are drawn from the mixed lot. The probability distribution of defective oranges is

MHT CET 2025 22nd April Evening Shift
26

The probability that a certain kind of component will survive a given test is $\frac{2}{3}$. The probability that at most 2 components out of 4 tested, will survive is

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27

A coin is tossed until one head appears or a tail appears 4 times in succession. The probability distribution of the number of tosses is

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28

The p.d.f. of a continuous random variable X is $f(x)=\left\{\begin{array}{cl}\frac{x^2}{18} & , \text { if }-3 < x < 3 \\ 0 & \text { otherwise }\end{array}\right.$

Then $\mathrm{P}[|\mathrm{X}|<2]=$

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29

In a single toss of a fair die, the odds against the event that number 4 or 5 turns up is

MHT CET 2025 22nd April Morning Shift
30

Numbers are selected at random, one at a time from the two-digit numbers $00,01,02,-------, 99$ with replacement. An event E occurs only if the product of the two digits of a selected number is 24. If four numbers are selected, then probability, that the event E occurs at least 3 times, is

MHT CET 2025 21st April Evening Shift
31

A random variable, $X$ has p.m.f. $\mathrm{P}(\mathrm{X}=x)=\frac{{ }^4 \mathrm{C}_x}{2^4}, x=0,1,2,3,4$ and $\mu$ and $\sigma^2$ are mean and variance respectively of random variable X , then

MHT CET 2025 21st April Evening Shift
32

$$ \text { The c.d.f. of a discrete random variable } \mathrm{X} \text { is } $$

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X} & -3 & -1 & 0 & 1 & 3 & 5 & 7 & 9 \\ \hline \mathrm{~F}(\mathrm{X}=x) & 0.1 & 0.3 & 0.5 & 0.65 & 0.75 & 0.85 & 0.90 & 1 \\ \hline \end{array} $$

Then $\frac{P[X=-3]}{P[X<0]}=$

MHT CET 2025 21st April Evening Shift
33

If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are mutually exclusive and exhaustive events of a sample space $S$ such that $P(B)=\frac{3}{2} P(A)$ and $P(C)=\frac{1}{2} P(B)$, then $P(A)=$

MHT CET 2025 21st April Evening Shift
34

If a random variable X follows the Binomial distribution $B(10, \quad p)$ such that $5 \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)$, then the value of $\frac{\mathrm{P}(\mathrm{X}=5)}{\mathrm{P}(\mathrm{X}=6)}$ is equal to

MHT CET 2025 21st April Morning Shift
35

The following is the probability distribution of X

$$ \begin{array}{|c|c|c|c|c|} \hline \mathrm{X} & 0 & 1 & 2 & 3 \\ \hline \mathrm{P}(\mathrm{X}=x) & \frac{1+\mathrm{p}}{5} & \frac{2-2 \mathrm{p}}{5} & \frac{2-\mathrm{p}}{5} & \frac{2 \mathrm{p}}{5} \\ \hline \end{array} $$

$$ \text { For a minimum value of } p \text {, the value of } 5 E(X) \text { is } $$

MHT CET 2025 21st April Morning Shift
36

A random variable X takes values $0,1,2,3$, ........ with probabilities. $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{2}\right)^x, \mathrm{k}$ is a constant, then $P(X=1)=$

MHT CET 2025 21st April Morning Shift
37

The probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two 'I's do not come together is

MHT CET 2025 21st April Morning Shift
38

Two numbers are selected at random, without replacement from the first 6 positive integers. Let $X$ denote the larger of the two numbers. Then $\mathrm{E}(\mathrm{X})=$

MHT CET 2025 20th April Evening Shift
39

For $\mathrm{k}=1,2,3$ the box $\mathrm{B}_{\mathrm{k}}$ contains k red balls and $(k+1)$ white balls. Let $P\left(B_1\right)=\frac{1}{2}, P\left(B_2\right)=\frac{1}{3}$ and $\mathrm{P}\left(\mathrm{B}_3\right)=\frac{1}{6} . \mathrm{A}$ box is selected at random and a ball is drawn from it. If a red ball is drawn from it, then the probability that it comes from box $\mathrm{B}_2$ is

MHT CET 2025 20th April Evening Shift
40

A random variable $X$ takes the values $0,1,2,3$, $\qquad$ with probability

$\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{5}\right)^x$, where k is a constant.

Then $\mathrm{P}(\mathrm{X}=0)$ is

MHT CET 2025 20th April Evening Shift
41

A fair coin is tossed 99 times. If X is the number of times head occur then $\mathrm{P}[\mathrm{X}=\mathrm{r}]$ is maximum when $\mathrm{r}=$

MHT CET 2025 20th April Evening Shift
42

If X is a binomial variable with range $\{0,1,2,3,4\}$ and $\mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=4)$ then the parameter ' $p$ ' of the binomial distribution is

MHT CET 2025 20th April Morning Shift
43

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. If X is the random variable of getting queens, then the value of $2 E(X)+3 E\left(X^2\right)$ for the number of queens is

MHT CET 2025 20th April Morning Shift
44

A random variable $X$ has the following probability distribution

$$ \begin{array}{|l|c|c|c|c|c|} \hline \mathrm{X}: & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(\mathrm{X}): & \mathrm{k} & 2 \mathrm{k} & 4 \mathrm{k} & 2 \mathrm{k} & \mathrm{k} \\ \hline \end{array} $$

then the value of $\mathrm{P}(1 \leqslant \mathrm{X}<4 \mid \mathrm{X} \leqslant 2)=$

MHT CET 2025 20th April Morning Shift
45

If two numbers $p$ and $q$ are chosen randomly from the set $\{1,2,3,4\}$, one by one, with replacement, then the probability of getting $\mathrm{p}^2 \geq 4 \mathrm{q}$ is

MHT CET 2025 20th April Morning Shift
46

If $X \sim B(n, p)$ then $\frac{P(X=k)}{P(X=k-1)}=$

MHT CET 2025 19th April Evening Shift
47

Let X be a discrete random variable. The probability distribution of X is given below

$$ \begin{array}{|c|c|c|c|} \hline \mathrm{X} & 30 & 10 & -10 \\ \hline \mathrm{P}(\mathrm{X}) & \frac{1}{5} & \mathrm{~A} & \mathrm{~B} \\ \hline \end{array} $$

and $\mathrm{E}(\mathrm{X})=4$, then the value of AB is equal to

MHT CET 2025 19th April Evening Shift
48

In a game, 3 coins are tossed. A person is paid $Rs \, 150$ if he gets all heads or all tails and he is supposed to pay ₹50 if he gets one head or two heads. The amount he can expect to win / lose on an average per game in ₹ is

MHT CET 2025 19th April Evening Shift
49

Let $A$ and $B$ are independent events with $\mathrm{P}(\mathrm{B})=\frac{2}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{11}{20}$, then $\mathrm{P}\left(\mathrm{A}^{\prime} \mid \mathrm{B}\right)$ is root of the equation

MHT CET 2025 19th April Evening Shift
50
A box contains 9 tickets numbered 1 to 9 both inclusive. If 3 tickets are drawn from the box one at a time, then the probability that they are alternatively either {odd, even, odd} or {even, odd, even} is
MHT CET 2025 19th April Morning Shift
51

The probability distribution of a discrete random variable X is

$\mathrm{X}$ 0 1 2 3 4
$\mathrm{P(X=}x)$ $\mathrm{2k}$ $\mathrm{k}$ $\mathrm{2k}$ $\mathrm{4k}$ $\mathrm{k}$

If $\mathrm{a}=\mathrm{P}(x<3)$ and $\mathrm{b}=\mathrm{P}(2 \leq \mathrm{X}<4)$, then

MHT CET 2025 19th April Morning Shift
52
If a random variable $X$ has the p.d.f. $f(x)=\left\{\begin{array}{cc}\frac{\mathrm{k}}{x^2+1} & , \text { if } 0< x< \infty \\ 0 & , \text { otherwise }\end{array}\right.$ then c.d.f. of X is
MHT CET 2025 19th April Morning Shift
53
If a random variable $X$ follows the Binomial distribution $\mathrm{B}(33, \mathrm{p})$ such that $3 \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)$, then the variance of X is
MHT CET 2025 19th April Morning Shift
54

If a discrete random variable X is defined as follows

$\mathrm{P}[\mathrm{X}=x]=\left\{\begin{array}{cl}\frac{\mathrm{k}(x+1)}{5^x}, & \text { if } x=0,1,2 \ldots \ldots . \\ 0, & \text { otherwise }\end{array}\right.$

then $\mathrm{k}=$

MHT CET 2024 16th May Evening Shift
55

Numbers are selected at random, one at a time from two digit numbers $10,11,12 \ldots ., 99$ with replacement. An event $E$ occurs if and only if the product of the two digits of a selected number is 18 . If four numbers are selected, then probability that the event E occurs at least 3 times is

MHT CET 2024 16th May Evening Shift
56

Two friends A and B apply for a job in the same company. The probabilities of A getting selected is $\frac{2}{5}$ and that of B is $\frac{4}{7}$. Then the probability, that one of them is selected, is

MHT CET 2024 16th May Evening Shift
57

If a random variable X has the following probability distribution values

$\mathrm{X}$ 0 1 2 3 4 5 6 7
$\mathrm{P(X):}$ 0 $\mathrm{k}$ $\mathrm{2k}$ $\mathrm{2k}$ $\mathrm{3k}$ $\mathrm{k^2}$ $\mathrm{2k^2}$ $\mathrm{7k^2+k}$

Then $P(X \geq 6)$ has the value

MHT CET 2024 16th May Morning Shift
58

A random variable X takes the values $0,1,2,3$ and its mean is 1.3 . If $\mathrm{P}(\mathrm{X}=3)=2 \mathrm{P}(\mathrm{X}=1)$ and $P(X=2)=0.3$, then $P(X=0)$ is

MHT CET 2024 16th May Morning Shift
59

Three persons $\mathrm{P}, \mathrm{Q}$ and R independently try to hit a target. If the probabilities of their hitting the target are $\frac{3}{4}, \frac{1}{2}$ and $\frac{5}{8}$ respectively, then the probability that the target is hit by P or Q but not by $R$, is

MHT CET 2024 16th May Morning Shift
60

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one by one, with replacement, then the variance of the number of green balls drawn is

MHT CET 2024 16th May Morning Shift
61
 

Suppose three coins are tossed simultaneously. If $X$ denotes the number of heads, then probability distribution of x is

MHT CET 2024 15th May Evening Shift
62

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five is

MHT CET 2024 15th May Evening Shift
63

If two fair dice are rolled, then the probability that the sum of the numbers on the upper faces is at least 9, is

MHT CET 2024 15th May Evening Shift
64

For the probability distribution

$x :$ 0 1 2 3 4 5
$p(x):$ $\mathrm{k}$ 0.3 0.15 0.15 0.1 2$\mathrm{k}$

The expected value of X is

MHT CET 2024 15th May Morning Shift
65

A random variable X has the following probability distribution

$X$ 1 2 3 4 5
$p(x)$ $\mathrm{k^2}$ $\mathrm{2k}$ $\mathrm{k}$ $\mathrm{2k}$ $\mathrm{5k^2}$

Then $\mathrm{p}(x \geq 2)$ is equal to

MHT CET 2024 15th May Morning Shift
66

The probability, that a year selected at random will have 53 Mondays, is

MHT CET 2024 15th May Morning Shift
67

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

MHT CET 2024 15th May Morning Shift
68

A random variable $X$ has the following probability distribution

$X=x$ 1 2 3 4 5 6 7 8
$P(X=x)$ 0.15 0.23 0.10 0.12 0.20 0.08 0.07 0.05

For the event $E=\{X$ is a prime number $\}$, $F=\{X<4\}$, then $P(E \cup F)$ is

MHT CET 2024 11th May Evening Shift
69

Let $\mathrm{A}, \mathrm{B}$ and C be three events, which are pairwise independent and $\bar{E}$ denote the complement of an event E . If $\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0$ and $\mathrm{P}(\mathrm{C})>0$, then $\mathrm{P}((\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) / C)$ is equal to

MHT CET 2024 11th May Evening Shift
70

A random variable x takes the values $0,1,2$, $3, \ldots$ with probability $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{5}\right)^x$, where k is a constant, then $\mathrm{P}(\mathrm{X}=0)$ is

MHT CET 2024 11th May Evening Shift
71

One hundred identical coins, each with probability p , of showing up heads are tossed once. If $0<\mathrm{p}<1$ and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of $p$ is

MHT CET 2024 11th May Evening Shift
72

The p.m.f. of a random variable X is given by

$$\begin{aligned} \mathrm{P}[\mathrm{X}=x] & =\frac{\binom{5}{x}}{2^5}, \text { if } x=0,1,2,3,4,5 \\ & =0, \text { otherwise } \end{aligned}$$

Then which of the following is not correct?

MHT CET 2024 11th May Morning Shift
73

If three fair coins are tossed, then variance of number of heads obtained, is

MHT CET 2024 11th May Morning Shift
74

If $A$ and $B$ are two independent events such that $\mathrm{P}\left(\mathrm{A}^{\prime}\right)=0.75, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.65$ and $\mathrm{P}(\mathrm{B})=\mathrm{p}$, then value of $p$ is

MHT CET 2024 11th May Morning Shift
75

The probability that a person who undergoes a bypass surgery will recover is 0.6 . the probability that of the six patients who undergo similar operations, half of them will recover is __________.

MHT CET 2024 11th May Morning Shift
76

$A$ and $B$ are independent events with $P(A)=\frac{3}{10}$, $\mathrm{P}(\mathrm{B})=\frac{2}{5}$, then $\mathrm{P}\left(\mathrm{A}^{\prime} \cup \mathrm{B}\right)$ has the value

MHT CET 2024 10th May Evening Shift
77

Minimum number of times a fair coin must be tossed, so that the probability of getting at least one head, is more than $99 \%$ is

MHT CET 2024 10th May Evening Shift
78

A random variable X assumes values $1,2,3, \ldots \ldots ., \mathrm{n}$ with equal probabilities. If $\operatorname{var}(X): E(X)=4: 1$, then $n$ is equal to

MHT CET 2024 10th May Evening Shift
79

In a game, 3 coins are tossed. A person is paid ₹ 100$, if he gets all heads or all tails; and he is supposed to pay ₹ 40 , if he gets one head or two heads. The amount he can expect to win/lose on an average per game in (₹) is

MHT CET 2024 10th May Morning Shift
80

In a Binomial distribution consisting of 5 independent trials, probabilities of exactly 1 and 2 successes are 0.4096 and 0.2048 respectively, then the probability, of getting exactly 4 successes, is

MHT CET 2024 10th May Morning Shift
81

A random variable X has the following probability distribution

$\mathrm{X}$ 1 2 3 4 5 6 7 8
$\mathrm{P(X=}x)$ 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05

For the events $\mathrm{E}=\{\mathrm{X}$ is prime number $\}$

$$\mathrm{F}=\{\mathrm{X}<4\}$$

Then $P(E \cup F)=$

MHT CET 2024 10th May Morning Shift
82
 

There are three events $\mathrm{A}, \mathrm{B}, \mathrm{C}$, one of which must and only one can happen. The odds are 8:3 against $\mathrm{A}, 5: 2$ against B and the odds against C is $43: 17 \mathrm{k}$, then value of k is

MHT CET 2024 10th May Morning Shift
83

Four persons can hit a target correctly with probabilities $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{5}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is

MHT CET 2024 9th May Evening Shift
84

If the mean and the variance of a Binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than one is equal to

MHT CET 2024 9th May Evening Shift
85

A bag contains 4 red and 3 black balls. One ball is drawn and then replaced in the bag and the process is repeated. Let X denote the number of times black ball is drawn in 3 draws. Assuming that at each draw each ball is equally likely to be selected, then probability distribution of $X$ is given by

MHT CET 2024 9th May Evening Shift
86

A service station manager sells gas at an average of ₹ 100 per hour on a rainy day, ₹ 150 per hour on a dubious day, ₹ 250 per hour on a fair day and ₹ $300$ on a clear sky. If weather bureau statistics show the probabilities of weather as follows, then his mathematical expectation is

Weather Clear Fair Dubious Rainy
Probability 0.50 0.30 0.15 0.05

MHT CET 2024 9th May Evening Shift
87

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability, that the three balls have different colours, is

MHT CET 2024 9th May Morning Shift
88

A random variable X takes values $-1,0,1,2$ with probabilities $\frac{1+3 \mathrm{p}}{4}, \frac{1-\mathrm{p}}{4}, \frac{1+2 \mathrm{p}}{4}, \frac{1-4 \mathrm{p}}{4}$ respectively, where p varies over $\mathbb{R}$. Then the minimum and maximum values of the mean of X are respectively.

MHT CET 2024 9th May Morning Shift
89

If the mean and the variance of Binomial variate $X$ are 2 and 1 respectively, then the probability that X takes a value greater than or equal to one is

MHT CET 2024 9th May Morning Shift
90

If $\mathrm{P}(\mathrm{X}=2)=0.3, \mathrm{P}(\mathrm{X}=3)=0.4, \mathrm{P}(\mathrm{X}=4)=0.3$, then the variance of random variable X is

MHT CET 2024 4th May Evening Shift
91

A man and his wife appear for an interview for two posts. The probability of the husband's selection is $\frac{1}{7}$ and that of the wife's selection is $\frac{1}{5}$. If they appear for the interview independently, then the probability that only one of them is selected, is

MHT CET 2024 4th May Evening Shift
92

The expected value of the sum of the two numbers obtained on the uppermost faces, when two fair dice are rolled, is

MHT CET 2024 4th May Evening Shift
93

For an entry to a certain course, a candidate is given twenty problems to solve. If the probability that the candidate can solve any problem is $\frac{3}{7}$, then the probability that he is unable to solve at most two problem is

MHT CET 2024 4th May Evening Shift
94

A random variable has the following probability distribution

$\mathrm{X:}$ 0 1 2 3 4 5 6 7
$\mathrm{P}(x):$ 0 $\mathrm{2p}$ $\mathrm{2p}$ $\mathrm{3p}$ $\mathrm{p^2}$ $\mathrm{2p^2}$ $\mathrm{7p^2}$ $\mathrm{2p}$

Then the value of p is

MHT CET 2024 4th May Morning Shift
95

Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that A or B occurs is $\frac{1}{2}$, then the probability of both of them occur together is

MHT CET 2024 4th May Morning Shift
96

A random variable $X$ has the following probability distribution

$\mathrm{X:}$ 1 2 3 4 5
$\mathrm{P(X):}$ $\mathrm{k^2}$ $\mathrm{2k}$ $\mathrm{k}$ $\mathrm{2k}$ $\mathrm{5k^2}$

Then $\mathrm{P(X > 2)}$ is equal to

MHT CET 2024 4th May Morning Shift
97

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability, that a student will get 4 or more correct answers just by guessing, is

MHT CET 2024 4th May Morning Shift
98

Let a random variable X have a Binomial distribution with mean 8 and variance 4 . If $\mathrm{P}(x \leqslant 2)=\frac{\mathrm{k}}{2^{16}}$, then k is equal to

MHT CET 2024 3rd May Evening Shift
99

For the probability distribution

$\mathrm{X:}$ $-2$ $-1$ $0$ $1$ $2$ $3$
$\mathrm{p}(x):$ 0.1 0.2 0.2 0.3 0.15 0.05

Then the $\operatorname{Var}(\mathrm{X})$ is

(Given : $$\left.(0.25)^2=0.0625,(0.35)^2=0.1225,(0.45)^2=0.2025\right)$$

MHT CET 2024 3rd May Evening Shift
100

Two cards are drawn successively with replacement from a well- shuffled pack of 52 cards. Let X denote the random variable of number of kings obtained in the two drawn cards. Then $\mathrm{P}(x=1)+\mathrm{P}(x=2)$ equals

MHT CET 2024 3rd May Evening Shift
101

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then mean of number of kings is

MHT CET 2024 3rd May Evening Shift
102

A random variable x has the following probability distribution. Then value of $k$ is _________ and $\mathrm{P}(3< x \leq 6)$ has the value

$\mathrm{X}=x$ 0 1 2 3 4 5 6 7 8
$\mathrm{P}(x)$ $\mathrm{k}$ $\mathrm{2k}$ $\mathrm{3k}$ $\mathrm{4k}$ $\mathrm{4k}$ $\mathrm{3k}$ $\mathrm{2k}$ $\mathrm{k}$ $\mathrm{k}$

MHT CET 2024 3rd May Morning Shift
103

Let $\mathrm{X} \sim \mathrm{B}\left(6, \frac{1}{2}\right)$, then $\mathrm{P}[|x-4| \leqslant 2]$ is

MHT CET 2024 3rd May Morning Shift
104

A person throws an unbiased die. If the number shown is even, he gains an amount equal to the number shown. If the number is odd, he loses an amount equal to the number shown. Then his expectation is ₹.

MHT CET 2024 3rd May Morning Shift
105

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Let X denote the random variable of number of jacks obtained in the two drawn cards. Then $P(X=1)+P(X=2)$ equals

MHT CET 2024 3rd May Morning Shift
106

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to

MHT CET 2024 2nd May Evening Shift
107

The probability distribution of a random variable X is given by

$\mathrm{X=}x_i$: 0 1 2 3 4
$\mathrm{P(X=}x_i)$ : 0.4 0.3 0.1 0.1 0.1

Then the variance of X is

MHT CET 2024 2nd May Evening Shift
108

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three persons apply for the same house is

MHT CET 2024 2nd May Evening Shift
109

A bag contains 4 Red and 6 Black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with 3 additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red is

MHT CET 2024 2nd May Morning Shift
110

If a discrete random variable X takes values $0,1,2,3, \ldots \ldots$. with probability $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1) 5^{-x}$, where k is a constant, then $\mathrm{P}(\mathrm{X}=0)$ is

MHT CET 2024 2nd May Morning Shift
111

Ten bulbs are drawn successively, with replacement, from a lot containing $10 \%$ defective bulbs, then the probability that there is at least one defective bulb, is

MHT CET 2024 2nd May Morning Shift
112

A fair die with numbers 1 to 6 on their faces is thrown. Let $$\mathrm{X}$$ denote the number of factors of the number, on the uppermost face, then the probability distribution of $$\mathrm{X}$$ is

MHT CET 2023 14th May Evening Shift
113

The p.m.f. of a random variable $$\mathrm{X}$$ is $$\mathrm{P}(x)=\left\{\begin{array}{cl}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \mathrm{n} \\ 0, & \text { otherwise }\end{array}\right.$$, then $$\mathrm{E}(\mathrm{X})$$ is

MHT CET 2023 14th May Evening Shift
114

There are 6 positive and 8 negative numbers. From these four numbers are chosen at random and multiplied. Then the probability, that the product is a negative number, is

MHT CET 2023 14th May Evening Shift
115

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs are selected at random from the lot and are sent to retail store. Then the probability that the store will receive at most one defective bulb is

MHT CET 2023 14th May Evening Shift
116

Two cards are drawn successively with replacement from well shuffled pack of 52 cards, then the probability distribution of number of queens is

MHT CET 2023 14th May Morning Shift
117

For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $$\frac{4}{5}$$, then the probability, that he is unable to solve less than two problems, is

MHT CET 2023 14th May Morning Shift
118

$$\text { If } f(x)= \begin{cases}3\left(1-2 x^2\right) & ; 0< x < 1 \\ 0 & ; \text { otherwise }\end{cases}$$ is a probability density function of $$\mathrm{X}$$, then $$\mathrm{P}\left(\frac{1}{4} < x < \frac{1}{3}\right)$$ is

MHT CET 2023 14th May Morning Shift
119

Three critics review a book. For the three critics the odds in favour of the book are $$2: 5, 3: 4$$ and $$4: 3$$ respectively. The probability that the majority is in favour of the book, is given by

MHT CET 2023 14th May Morning Shift
120

Two dice are rolled. If both dice have six faces numbered $$1,2,3,5,7,11$$, then the probability that the sum of the numbers on upper most face is prime, is

MHT CET 2023 13th May Evening Shift
121

A random variable $$X$$ has the probability distribution

$$X=x$$ 1 2 3 4 5 6 7 8
$$P(X=x)$$ 0.15 0.23 0.12 0.20 0.08 0.10 0.05 0.07

For the events $$E=\{X$$ is a prime number $$\}$$ and $$F=\{x<5\}, P(E U F)$$ is

MHT CET 2023 13th May Evening Shift
122

A random variable $$X$$ has the following probability distribution

$$\mathrm{X}=x$$ 0 1 2
$$\mathrm{P(X}=x)$$ $$\mathrm{4k-10k^2}$$ $$\mathrm{5k-1}$$ $$\mathrm{3k^3}$$

then P(X < 2) is

MHT CET 2023 13th May Morning Shift
123

Let $$\mathrm{X}$$ be random variable having Binomial distribution $$B(7, p)$$. If $$P[X=3]=5 P[X=4]$$, then variance of $$\mathrm{X}$$ is

MHT CET 2023 13th May Morning Shift
124

If a continuous random variable $$\mathrm{X}$$ has probability density function $$\mathrm{f}(x)$$ given by

$$f(x)=\left\{\begin{array}{cl} a x & , \text { if } 0 \leq x<1 \\ a & , \text { if } 1 \leq x<2 \\ 3 a-a x & , \text { if } 2 \leq x \leq 3 \\ 0 & , \text { otherwise } \end{array}\right.$$,

then a has the value

MHT CET 2023 13th May Morning Shift
125

A card is drawn at random from a well shuffled pack of 52 cards. The probability that it is black card or face card is

MHT CET 2023 13th May Morning Shift
126

An irregular six faced die is thrown and the probability that, in 5 throws it will give 3 even numbers is twice the probability that it will give 2 even numbers. The number of times, in 6804 sets of 5 throws, you expect to give no even number is

MHT CET 2023 12th May Evening Shift
127

A box contains 100 tickets numbered 1 to 100 . A ticket is drawn at random from the box. Then the probability, that number on the ticket is a perfect square, is

MHT CET 2023 12th May Evening Shift
128

Three fair coins with faces numbered 1 and 0 are tossed simultaneously. Then variance (X) of the probability distribution of random variable $$\mathrm{X}$$, where $$\mathrm{X}$$ is the sum of numbers on the upper most faces, is

MHT CET 2023 12th May Evening Shift
129

The p.m.f of random variate $$\mathrm{X}$$ is $$P(X)= \begin{cases}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \ldots, \mathrm{n} \\ 0, & \text { otherwise }\end{cases}$$

Then $$\mathrm{E}(\mathrm{X})=$$

MHT CET 2023 12th May Morning Shift
130

An experiment succeeds twice as often as it fails. Then the probability, that in the next 6 trials there will be atleast 4 successes, is

MHT CET 2023 12th May Morning Shift
131

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then the probability distribution of number of jacks is

MHT CET 2023 12th May Morning Shift
132

$$\mathrm{A}$$ and $$\mathrm{B}$$ are independent events with $$\mathrm{P}(\mathrm{A})=\frac{1}{4}$$ and $$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=2 \mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})$$, then $$\mathrm{P}(\mathrm{B})$$ is

MHT CET 2023 12th May Morning Shift
133

Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Then mean of number of tens is

MHT CET 2023 11th May Evening Shift
134

A fair die is tossed twice in succession. If $$\mathrm{X}$$ denotes the number of fours in two tosses, then the probability distribution of $$\mathrm{X}$$ is given by

MHT CET 2023 11th May Evening Shift
135

If $$\mathrm{A}$$ and $$\mathrm{B}$$ are two events such that $$\mathrm{P}(\mathrm{A})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{1}{3}$$, then the value of $$\mathrm{P}\left(\mathrm{A}^{\prime} / \mathrm{B}^{\prime}\right)+\mathrm{P}\left(\mathrm{B}^{\prime} / \mathrm{A}^{\prime}\right)$$ is

MHT CET 2023 11th May Evening Shift
136

Let a random variable $$\mathrm{X}$$ have a Binomial distribution with mean 8 and variance 4. If $$\mathrm{P}(\mathrm{X} \leq 2)=\frac{\mathrm{K}}{2^{16}}$$, then $$\mathrm{K}$$ is

MHT CET 2023 11th May Evening Shift
137

From a lot of 20 baskets, which includes 6 defective baskets, a sample of 2 baskets is drawn at random one by one without replacement. The expected value of number of defective basket is

MHT CET 2023 11th May Morning Shift
138

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with these three vertices is equilateral, equals ___________.

MHT CET 2023 11th May Morning Shift
139

A binomial random variable $$\mathrm{X}$$ satisfies $$9. p(X=4)=p(X=2)$$ when $$n=6$$. Then $$p$$ is equal to

MHT CET 2023 11th May Morning Shift
140

The three ships namely A, B and C sail from India to Africa. If the odds in favour of the ships reaching safely are $$2: 5,3: 7$$ and $$6: 11$$ respectively, then probability of all of them arriving safely is

MHT CET 2023 10th May Evening Shift
141

If the sum of mean and variance of a Binomial Distribution is $$\frac{15}{2}$$ for 10 trials, then the variance is

MHT CET 2023 10th May Evening Shift
142

In a game, 3 coins are tossed. A person is paid ₹ 7 /-, if he gets all heads or all tails; and he is supposed to pay ₹ 3 /-, if he gets one head or two heads. The amount he can expect to win on an average per game is ₹

MHT CET 2023 10th May Evening Shift
143

A fair die is tossed twice in succession. If $$\mathrm{X}$$ denotes the number of sixes in two tosses, then the probability distribution of $$\mathrm{X}$$ is given by

MHT CET 2023 10th May Evening Shift
144

For a binomial variate $$\mathrm{X}$$ with $$\mathrm{n}=6$$ if $$P(X=4)=\frac{135}{2^{12}}$$, then its variance is

MHT CET 2023 10th May Morning Shift
145

The p.d.f. of a discrete random variable is defined as $$\mathrm{f}(x)=\left\{\begin{array}{l} \mathrm{k} x^2, 0 \leq x \leq 6 \\ 0, \text { otherwise } \end{array}\right.$$

Then the value of $$F(4)$$ (c.d.f) is

MHT CET 2023 10th May Morning Shift
146

A player tosses 2 fair coins. He wins ₹5 if 2 heads appear, ₹ 2 if one head appears and ₹ 1 if no head appears. Then the variance of his winning amount in ₹ is :

MHT CET 2023 10th May Morning Shift
147

Three critics review a book. For the three critics the odds in favor of the book are $$2: 5, 3: 4$$ and $$4: 3$$ respectively. The probability that the majority is in favor of the book, is given by

MHT CET 2023 10th May Morning Shift
148

A man takes a step forward with probability 0.4 and backwards with probability 0.6 . The probability that at the end of eleven steps, he is one step away from the starting point is

MHT CET 2023 9th May Evening Shift
149

A problem in statistics is given to three students A, B and C. Their probabilities of solving the problem are $$\frac{1}{2}, \frac{1}{3}$$ and $$\frac{1}{4}$$ respectively. If all of them try independently, then the probability, that problem is solved, is

MHT CET 2023 9th May Evening Shift
150

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards, then mean of number of queens is

MHT CET 2023 9th May Evening Shift
151

In a Binomial distribution with $$\mathrm{n}=4$$, if $$2 \mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=2)$$, then the variance is

MHT CET 2023 9th May Morning Shift
152

$$\mathrm{A}, \mathrm{B}, \mathrm{C}$$ are three events, one of which must and only one can happen. The odds in favor of $$\mathrm{A}$$ are $$4: 6$$, the odds against $$B$$ are $$7: 3$$. Thus, odds against $$\mathrm{C}$$ are

MHT CET 2023 9th May Morning Shift
153

The probability mass function of random variable X is given by

$$P[X=r]=\left\{\begin{array}{ll} \frac{{ }^n C_r}{32}, & n, r \in \mathbb{N} \\ 0, & \text { otherwise } \end{array} \text {, then } P[X \leq 2]=\right.$$

MHT CET 2023 9th May Morning Shift
154

Three fair coins numbered 1 and 0 are tossed simultaneously. Then variance Var (X) of the probability distribution of random variable $$\mathrm{X}$$, where $$\mathrm{X}$$ is the sum of numbers on the uppermost faces, is

MHT CET 2023 9th May Morning Shift
155

The incidence of occupational disease in an industry is such that the workmen have a $$10 \%$$ chance of suffering from it. The probability that out of 5 workmen, 3 or more will contract the disease is

MHT CET 2022 11th August Evening Shift
156

If $$P(A \cup B)=0.7, P(A \cap B)=0.2$$, then $$P\left(A^{\prime}\right)+P\left(B^{\prime}\right)$$ is

MHT CET 2022 11th August Evening Shift
157

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then mean of number of kings is

MHT CET 2022 11th August Evening Shift
158

The probability that at least one of the events $$E_1$$ and $$E_2$$ occurs is 0.6. If the simultaneous occurrence of $$\mathrm{E}_1$$ and $$\mathrm{E}_2$$ is $$0.2, \mathrm{P}\left(\mathrm{E}_1^{\prime}\right)+\mathrm{P}\left(\mathrm{E}_2^{\prime}\right)=$$

MHT CET 2021 24th September Evening Shift
159

Two dice are thrown simultaneously. If X denotes the number of sixes, then the expectation of X is

MHT CET 2021 24th September Evening Shift
160

The probability distribution of a random variable X is

$$\mathrm{X=x}$$ 1 2 3 ......... $$\mathrm{n}$$
$$\mathrm{P(X=x)}$$ $$\mathrm{\frac{1}{n}}$$ $$\mathrm{\frac{1}{n}}$$ $$\mathrm{\frac{1}{n}}$$ ......... $$\mathrm{\frac{1}{n}}$$

then Var(X) =

MHT CET 2021 24th September Evening Shift
161

A fair coin is tossed for a fixed number of times. If probability of getting 7 heads is equal to probability of getting 9 heads, then probability of getting 2 heads is

MHT CET 2021 24th September Evening Shift
162

If the probability distribution function of a random variable X is given as

$$\mathrm{X=x_i}$$ $$-2$$ $$-1$$ 0 1 2
$$\mathrm{P(X=x_i)}$$ 0.2 0.3 0.15 0.25 0.1

Then F(0) is equal to

MHT CET 2021 24th September Morning Shift
163

If $$\mathrm{P}(\mathrm{A})=\frac{3}{10}, \mathrm{P}(\mathrm{B})=\frac{2}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{5}$$, then $$\mathrm{P}(\mathrm{A} / \mathrm{B}) \times \mathrm{P}(\mathrm{B} / \mathrm{A})=$$

MHT CET 2021 24th September Morning Shift
164

The probability distribution of a discrete random variable X is

$$\mathrm{X}$$ 1 2 3 4 5 6
$$\mathrm{P(X)}$$ K 2K 3K 4K 5K 6K

Find the value of $$\mathrm{P}(2<\mathrm{X}<6)$$

MHT CET 2021 24th September Morning Shift
165

A die is thrown four times. The probability of getting perfect square in at least one throw is

MHT CET 2021 24th September Morning Shift
166

A man is known to speck truth 3 out of 4 times. He throws a die and reports that it is 6. Then the probability that it is actually 6 is

MHT CET 2021 23rd September Evening Shift
167

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is

MHT CET 2021 23rd September Evening Shift
168

Let two cards are drawn at random from a pack of 52 playing cards. Let X be the number of aces obtained. Then the value of E(X) is

MHT CET 2021 23rd September Evening Shift
169

A fair coin is tossed 100 times. The probability of getting a head for even number of times is

MHT CET 2021 23rd September Evening Shift
170

If the function defined by $$f(x)=K(x-x^2)$$ if $$0 < x < 1=0$$, otherwise is the p.d.f. of a r.v.X, then the value of $$P\left(X<\frac{1}{2}\right)$$ is

MHT CET 2021 23rd September Evening Shift
171

The probability distribution of the number of doublets in four throws of a pair of dice is given by

MHT CET 2021 23th September Morning Shift
172

For the probability distribution given by following

$$\mathrm{x}$$ 5 6 7 8 9 10 11
$$\mathrm{P(X=x)}$$ 0.07 0.2 0.3 $$\mathrm{k}$$ 0.07 0.04 0.02

Var(X) =

MHT CET 2021 23th September Morning Shift
173

A random variable X has the following probability distribution

$$x$$ 0 1 2 3 4 5 6 7 8
$$P(X=x)$$ K 2K 3K 4K 4K 3K 2K K K

Then $$\mathrm{P}(3<\mathrm{x} \leq 6)=$$

MHT CET 2021 23th September Morning Shift
174

Rooms in a hotel are numbered from 1 to 19. Rooms are allocated at random as guests arrive. The first guest to arrive is given a room which is a prime number. The probability that the second guest to arrive is given a room which is a prime number is

MHT CET 2021 23th September Morning Shift
175

The distribution function $$F(X)$$ of discrete random variable $$X$$ is given by

$$\mathrm{X}$$ 1 2 3 4 5 6
$$\mathrm{F (X=x)}$$ 0.2 0.37 0.48 0.62 0.85 1

Then $$\mathrm{P[X=4]+P[x=5]=}$$

MHT CET 2021 22th September Evening Shift
176

First bag contains 3 red and 5 black balls and second bag contains 6 red and 4 black balls. A ball is drawn from each bag. The probability that one ball is red and the other is black, is

MHT CET 2021 22th September Evening Shift
177

A fair coin is tossed 4 times. If $$X$$ is a random variable which indicates number of heads, then $$\mathrm{P}[\mathrm{X}<3]=$$

MHT CET 2021 22th September Evening Shift
178

If the mean and variance of a binomial distribution are 4 and 2 respectively, then probability of getting 2 heads is

MHT CET 2021 22th September Evening Shift
179

For two events $$\mathrm{A}$$ and $$\mathrm{B}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5}{6}, \mathrm{P}(\mathrm{A})=\frac{1}{6}, \mathrm{P}(\mathrm{B})=\frac{2}{3}$$, then $$\mathrm{A}$$ and $$\mathrm{B}$$ are

MHT CET 2021 22th September Morning Shift
180

A random variable X has following distribution

$$\mathrm{X = x}$$ 1 2 3 4 5 6
$$\mathrm{P(X = x)}$$ k 3k 5k 7k 8k k

Then P (2 $$\le$$ x < 5) =

MHT CET 2021 22th September Morning Shift
181

A coin is tossed three times. If X denotes the absolute difference between the number of heads and the number of tails, then P (X = 1) =

MHT CET 2021 22th September Morning Shift
182

A random variable X $$\sim$$ B (n, p), if values of mean and variance of X are 18 and 12 respectively, then n =

MHT CET 2021 22th September Morning Shift
183

an urn contains 9 balls of which 3 are red, 4 are blue and 2 are green. Three balls are drawn at random from the urn. The probability that the three balls have difference colours is

MHT CET 2021 21th September Evening Shift
184

It is observed that $$25 \%$$ of the cases related to child labour reported to the police station are solved. If 6 new cases are reported, then the probability that at least 5 of them will be solved is

MHT CET 2021 21th September Evening Shift
185

In a meeting $$60 \%$$ of the members favour and $$40 \%$$ oppose a certain proposal. A member is selected at random and we take $$\mathrm{X}=0$$ if he opposed and $$\mathrm{X}=1$$ if he is in favour, then $$\operatorname{Var} \mathrm{X}=$$

MHT CET 2021 21th September Morning Shift
186

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs selected at random from the lot and sent to retain store, then the probability that the store will receive at most one defective bulb is

MHT CET 2021 21th September Morning Shift
187

A coin is tossed and a die is thrown. The probability that the outcome will be head or a number greater than 4 or both, is

MHT CET 2021 21th September Morning Shift
188

If $$\mathrm{X}$$ is a random variable with p.m.f. as follows.

$$\begin{aligned} \mathrm{P}(\mathrm{X}=\mathrm{x}) & =\frac{5}{16}, \mathrm{x}=0,1 \\ & =\frac{\mathrm{kx}}{48}, \mathrm{x}=2, \quad \text { then } \mathrm{E}(\mathrm{x})= \\ & =\frac{1}{4}, \mathrm{x}=3 \end{aligned}$$

MHT CET 2021 21th September Morning Shift
189

The p.m.f. of a random variable X is $$\mathrm{P(X = x) = {1 \over {{2^5}}}\left( {_x^5} \right),x = 0,1,2,3,4,5}=0$$ then

MHT CET 2021 20th September Evening Shift
190

The variance of the following probability distribution is,

MHT CET 2021 20th September Evening Shift Mathematics - Probability Question 212 English

MHT CET 2021 20th September Evening Shift
191

If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, then probability of a success is

MHT CET 2021 20th September Evening Shift
192

Two unbiased dice are thrown. Then the probability that neither a doublet nor a total of 10 will appear is

MHT CET 2021 20th September Evening Shift
193

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the dice is a prime number, is

MHT CET 2021 20th September Morning Shift
194

A random variable X has the following probability distribution

$$\mathrm{X=x}$$ 0 1 2 3 4 5 6 7
$$\mathrm{P[X=x]}$$ 0 $$\mathrm{k}$$ $$\mathrm{2k}$$ $$\mathrm{2k}$$ $$\mathrm{3k}$$ $$\mathrm{k^2}$$ $$\mathrm{2k^2}$$ $$\mathrm{7k^2+k}$$

then F(4) =

MHT CET 2021 20th September Morning Shift
195

Rajesh has just bought a VCR from Maharashtra Electronics and the shop offers after sales service contract for Rs. 1000 for the next five years. Considering the experience of VCR users, the following distribution of maintenance expenses for the next five years is formed.

Expenses 0 500 1000 1500 2000 2500 3000
Probability 0.35 0.25 0.15 0.10 0.08 0.05 0.02

The expected value of maintenance cost is :

MHT CET 2021 20th September Morning Shift
196

If $$X \sim B(4, p)$$ and $$P(X=0)=\frac{16}{81}$$, then $$P(X=4)=$$

MHT CET 2021 20th September Morning Shift
197

The odds in favour of getting sum multiple of 3 , when pair of dice are thrown is

MHT CET 2020 19th October Evening Shift
198

If $X$ is a.r.v. with c.d.f $F(x)$ and its probability distribution is given by

$X=x$ $-1.5$ $-0.5$ 0.5 1.5 2.5
$P(X=x)$ 0.05 0.2 0.15 0.25 0.35

then, $F(1.5)-F(-0.5)=$

MHT CET 2020 19th October Evening Shift
199

The odds in favour of drawing a king from a pack of 52 playing cards is

MHT CET 2020 16th October Evening Shift
200

Out of 100 people selected at random, 10 have common cold. If five persons selected at random from the group, then the probability that at most one person will have common cold is

MHT CET 2020 16th October Evening Shift
201

The pdf of a continuous r.v. $$X$$ is given by $$f(x)=\frac{x}{8}, 0 < x < 4=0$$, otherwise, then $$P(X \leq 2)$$ is

MHT CET 2020 16th October Evening Shift
202

If a die is thrown at random, then the expectation of the number on it is

MHT CET 2020 16th October Evening Shift
203

The probability that bomb will miss the target is 0.2. Then, the probability that out of 10 bombs dropped exactly 2 will hit the target is

MHT CET 2020 16th October Morning Shift
204

The letters of the word 'LOGARITHM' are arranged at random. The probability that arrangement starts with vowel and end with consonant is

MHT CET 2020 16th October Morning Shift
205

The p.d.f of c.r.v $$X$$ is given by $$f(x)=\frac{x+2}{18}$$, if $$-2

MHT CET 2020 16th October Morning Shift
206

If the p.m.f of a. r.v. $$X$$ is given by

$$P(X=x)=\frac{{ }^5 C_x}{2^5}$$

if $$x=0,1,2, \ldots \ldots . .5=0$$,

0 , otherwise,

then which of the following is not true?

MHT CET 2020 16th October Morning Shift
207

Let $X$ be the number of successes in ' $n$ ' independent Bernoulli trials with probability of success $p=\frac{3}{4}$. The least value of ' $n$ ' so that $P(X \geq 1) \geq 0.9375$ is ......

MHT CET 2019 3rd May Morning Shift
208

The probability that three cards drawn from a pack of 52 cards, all are red is

MHT CET 2019 3rd May Morning Shift
209

$$\begin{aligned} &\text { The pdf of a random variable } X \text { is }\\ &\begin{aligned} f(x) & =3\left(1-2 x^2\right), & & 0< x<1 \\ & =0 & & \text { otherwise } \end{aligned} \end{aligned}$$

The $P\left(\frac{1}{4}< x<\frac{1}{3}\right)=\ldots$

MHT CET 2019 3rd May Morning Shift
210

A player tosses 2 fair coins. He wins Rs. 5 if 2 heads appear, Rs. 2 If 1 head appear and Rs. 1 if no head appears, then variance of his winning amount is

MHT CET 2019 3rd May Morning Shift
211

In a bionomial distribution, mean is 18 and variance is 12 then $p=$ ...........

MHT CET 2019 2nd May Evening Shift
212

The p.d.f of a random variable $x$ is given by

$$\begin{aligned} & f(x)=\frac{1}{4 a}, \quad 00) \\ & =0 \text {, otherwise } \end{aligned}$$

and $P\left(x<\frac{3 a}{2}\right)=k P\left(x>\frac{5 a}{2}\right)$ then $k=$ ..............

MHT CET 2019 2nd May Evening Shift
213

If three dices are thrown then the probability that the sum of the numbers on their uppermost faces to be atleast 5 is

MHT CET 2019 2nd May Evening Shift
214

It is observed that $25 \%$ of the cases related to child labour reported to the police station are solved. If 6 new cases are reported, then the probability that atleast 5 of them will be solved is

MHT CET 2019 2nd May Morning Shift
215

A bag contains 6 white and 4 black balls. Two balls are drawn at random. The probability that they are of the same colour is ...........

MHT CET 2019 2nd May Morning Shift
216

A random variable X has following probability distribution

$X=x$ 1 2 3 4 5 6
$P(X=x)$ K 3K 5K 7K 8K K

Then $P(2 \leq X<5)=\ldots \ldots$

MHT CET 2019 2nd May Morning Shift
217

If the c.d.f (cumulative distribution function) is given by $F(x)=\frac{x-25}{10}$, then $P(27 \leq x \leq 33)=\ldots \ldots$

MHT CET 2019 2nd May Morning Shift
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