1
MCQ (Single Correct Answer)

JEE Main 2016 (Online) 10th April Morning Slot

An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is :
A
$${{240} \over {729}}$$
B
$${{192} \over {729}}$$
C
$${{256} \over {729}}$$
D
$${{496} \over {729}}$$

Explanation

Probability of fail, P(F) = p

$$ \therefore $$   Probability of success, P(S) = 2p

We know,

p + 2Pp = 1

$$ \Rightarrow $$   p = $${1 \over 3}$$

$$ \therefore $$   Now, probability of at least 5 success in 6 trials,

P(x $$ \ge $$ 5)

= P(x = 5) + P (x = 6)

= 6C5 $${\left( {{2 \over 3}} \right)^5}{\left( {{1 \over 3}} \right)^1}{ + ^6}{C_6}{\left( {{2 \over 3}} \right)^6}{\left( {{1 \over 3}} \right)^o}$$

= $${\left( {{2 \over 3}} \right)^5}\left( {{6 \over 3} + {2 \over 3}} \right)$$

= $${{256} \over {729}}$$
2
MCQ (Single Correct Answer)

JEE Main 2017 (Offline)

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:
A
6
B
4
C
$${6 \over {25}}$$
D
$${{12} \over 5}$$

Explanation

We can apply binomial probability distribution

n = 10

p = Probability of drawing a green ball = $${{15} \over {25}}$$ = $${3 \over 5}$$

Also q = 1 - $${3 \over 5}$$ = $${2 \over 5}$$

Variance = npq

= $$10 \times {3 \over 5} \times {2 \over 5}$$ = $${{12} \over 5}$$
3
MCQ (Single Correct Answer)

JEE Main 2017 (Offline)

If two different numbers are taken from the set {0, 1, 2, 3, ........, 10}; then the probability that their sum as well as absolute difference are both multiple of 4, is:
A
$${{12} \over {55}}$$
B
$${{14} \over {45}}$$
C
$${{7} \over {55}}$$
D
$${{6} \over {55}}$$

Explanation

Let A = {0, 1, 2, 3, 4, ......., 10}

Total number of ways of selecting 2 different numbers from A is

n (S) = 11C2 = 55, where 'S' denotes sample space

Let E be the given event

$$ \therefore $$ E = {(0, 4), (0, 8), (2, 6), (2, 10), (4, 8), (6, 10)}

$$ \Rightarrow $$ n (E) = 6

$$ \therefore $$ P(E) = $${{n\left( E \right)} \over {n\left( S \right)}}$$ = $${6 \over {55}}$$
4
MCQ (Single Correct Answer)

JEE Main 2017 (Offline)

For three events A, B and C,

P(Exactly one of A or B occurs)
= P(Exactly one of B or C occurs)
= P (Exactly one of C or A occurs) = $${1 \over 4}$$
and P(All the three events occur simultaneously) = $${1 \over {16}}$$.

Then the probability that at least one of the events occurs, is:
A
$${7 \over {16}}$$
B
$${7 \over {64}}$$
C
$${3 \over {16}}$$
D
$${7 \over {32}}$$

Explanation

Given, P (A $$ \cap $$ B $$ \cap $$ C) = $${1 \over {16}}$$

P (exactly one of A or B occurs)

= P(A) + P (B) – 2P (A $$ \cap $$ B) = $${1 \over 4}$$ .....(1)

P (Exactly one of B or C occurs)

= P(B) + P (C) – 2P (B $$ \cap $$ C) = $${1 \over 4}$$ .....(2)

P (Exactly one of C or A occurs)

= P(C) + P(A) – 2P (C $$ \cap $$ A) = $${1 \over 4}$$ .....(3)

Adding (1), (2) and (3),we get

2[ P(A) + P(B) + P (C) - P (A $$ \cap $$ B)

- P (B $$ \cap $$ C) - P (C $$ \cap $$ A)] = $${3 \over 4}$$

$$ \Rightarrow $$ P(A) + P(B) + P (C) - P (A $$ \cap $$ B)

- P (B $$ \cap $$ C) - P (C $$ \cap $$ A) = $${3 \over 8}$$

$$ \therefore $$ P(atleast one event occurs)

= P (A $$ \cup $$ B $$ \cup $$ C)

= P(A) + P(B) + P (C) - P (A $$ \cap $$ B)

- P (B $$ \cap $$ C) - P (C $$ \cap $$ A) + P (A $$ \cap $$ B $$ \cap $$ C)

= $${3 \over 8} + {1 \over {16}}$$ = $${7 \over {16}}$$

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