Probability — Class 12 Mathematics PYQs

If $E$ and $F$ are two independent events such that $P(E)=\frac{2}{3}$, $P(F)=\frac{3}{7}$, then $P(E|\bar{F})$ is equal to :

(A) $\frac{1}{6}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) $\frac{7}{9}$

If $E$ and $F$ are two events such that $P(E)>0$ and $P(F)\ne 1$, then $P(\bar{E}|\bar{F})$ is

(A) $\frac{P(\bar{E})}{P(\bar{F})}$
(B) $1-P(\bar{E}|F)$
(C) $1-P(E|F)$
(D) $\frac{1-P(E\cup F)}{P(\bar{F})}$

In the following probability distribution, the value of $p$ is :

$X$	0	1	2	3
$P(X)$	$p$	$p$	$0.3$	$2p$

(A) $\frac{7}{40}$
(B) $\frac{1}{10}$
(C) $\frac{9}{35}$
(D) $\frac{1}{4}$

A random variable $X$ has the probability distribution given below :

$X$	$-4$	$-3$	$-2$	$-1$	$0$
$P(X)$	$0.1$	$0.2$	$0.3$	$0.2$	$0.2$

Then $E(X)$ of distribution is

(A) $-1.8$
(B) $-1$
(C) $1$
(D) $1.8$

(a) The probability distribution for the number of students being absent in a class on a Saturday is as follows :

$X$	0	2	4	5
$P(X)$	$p$	$2p$	$3p$	$p$

Where $X$ is the number of students absent.

(i) Calculate $p$.

(ii) Calculate the mean of the number of absent students on Saturday.

OR

(b) For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and others were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.

(a) A die with numbers 1 to 6 is biased such that $P(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.

OR

(b) Two dice are thrown. Defined are the following two events $A$ and $B$ :
$A=\{(x,y):x+y=9\}$, $B=\{(x,y):x\ne 3\}$, where $(x,y)$ denote a point in the sample space.
Check if events $A$ and $B$ are independent or mutually exclusive.

A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively.

Based on the above information, answer the following :

(i) What is the probability that a customer after availing the loan will default on the loan repayment ?

(ii) A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest ?

Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated).

Based on the above, answer the following :

(i) (a) What is the probability that a randomly selected car is an electric car ?

OR

(i) (b) What is the probability that a randomly selected car is a petrol car ?

(ii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet ?

(iii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi ?

Let $E$ be an event of a sample space $S$ of an experiment, then $P(S|E)$ = :

If $P(A|B)=P(A^{\prime }|B)$, then which of the following statements is true?