Probability · Mathematics · Class 12

If $A$ and $B$ are events such that $P(A / B)=P(B / A) \neq 0$, then

If for any two events $$\mathrm{A}$$ and $$\mathrm{B}, P(A)=\frac{4}{5}$$ and $$P(A \cap B)=\frac{7}{10}$$, then $$P(B / A)$$ is equals to:

Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is:

Assertion (A): Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is $$\frac{1}{3}$$.

Reason (R): Let E and F be two events with a random experiment, then $$\mathrm{P}(\mathrm{F} / \mathrm{E})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{F})}{\mathrm{P}(\mathrm{E})}$$.

The random variable $X$ has the following probability distribution where $a$ and $b$ are some constants:

If the mean $E(X)=3$, then find values of $a$ and $b$ and hence determine $P(X \geq 3)$.

A departmental store sends bills to charge its customers once a month. Past experience shows that $70 \%$ of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month.

Based on the above information, answer the following questions:

(i) Let $E_1$ and $E_2$ respectively, denote the event of customer paying or not paying the first month bill in time.

(ii) Let A denotes the event of customer paying second month's bill in time, then find $P\left(A \mid E_1\right)$ and $P\left(A \mid E_2\right)$.

(iii) Find the probability of customer paying second month's bill in time.

OR

(iii) Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.

(a) In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $$\frac{3}{5}$$ be the probability that he knows the answer and $$\frac{2}{5}$$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $$\frac{1}{3}$$. What is the probability that the student knows the answer, given that he answered it correctly?

OR

(b) A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize.