Probability: CBSE Class 12 Previous Year Questions

Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is $\frac{2}{3}$.
Reason (R): For any two events A and B, $P(A|B)=\frac{P(A\cup B)}{P(B)}$

1. (A)Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true and Reason (R) is false.
4. (D)Assertion (A) is false and Reason (R) is true.

For two events A and B such that $P(A)\ne 0$ and $P(B)\ne 1$, $P(A^{\prime }/B^{\prime })=$

1. (A)$1-P(A/B)$
2. (B)$1-P(A^{\prime }/B)$
3. (C)$\frac{1-P(A\cap B)}{P(B^{\prime })}$
4. (D)$\frac{1-P(A\cup B)}{P(B^{\prime })}$

If E and F are two independent events such that $P(E)=\frac{3}{10}$, $P(E\cup F)=\frac{1}{2}$, then $P(E|F)-P(F|E)$ is equal to :

1. (A)$\frac{2}{7}$
2. (B)$\frac{3}{35}$
3. (C)$\frac{1}{70}$
4. (D)$\frac{1}{7}$

If $3P(A)=P(B)=\frac{3}{5}$ and $P(A|B)=\frac{2}{3}$, then $P(A\cup B)$ is :

1. (A)$\frac{3}{5}$
2. (B)$\frac{1}{5}$
3. (C)$\frac{2}{15}$
4. (D)$\frac{2}{5}$

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

The probability of simultaneous occurrence of atleast one of the two events X and Y is a. If the probability that exactly one of the events X, Y occurs is b, prove that $P(X^{\prime })+P(Y^{\prime })=2-2a+b$.

The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed.
The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that
(i) target is hit
(ii) atleast one shot misses the target.

Mother, Father and Son line up at random for a family picture. Let events E : Son on one end and F : Father in the middle. Find P(E/F).

A survey was conducted on the patients who have undergone knee replacement surgeries.
It was found that, Robotic Knee replacement surgeries have 90% success rate.
On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that :
(i) exactly one surgery is successful,
(ii) at most two surgeries are successful.

In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.