Probability - ISC Class 12 Mathematics Questions with Answers, Page 3

Rahul, a class XII student, has a probability of getting a grade A in the examination of three subjects namely Mathematics, Physics and Chemistry are $0.2$, $0.3$ and $0.5$ respectively. Find the probability that he gets a grade A in none of the subjects.

In a school, there are 30 teachers in the examination committee. Out of these, 20 never commit any error in their work. Two teachers are selected at random from the committee. The random variable $X$ is the number of selected teachers who never make an error in their work.

Naman has to attend his friend's marriage. He can reach the wedding venue by train, bus or a two-wheeler. The probability of Naman using these as means of transport are $\frac{3}{10}$, $\frac{1}{5}$ and $\frac{1}{2}$ respectively. The probability of Naman reaching the wedding venue on time are $\frac{1}{4}$, $\frac{1}{3}$ and $\frac{2}{5}$ if he uses train, bus and two-wheeler respectively. Naman reached the venue late. What is the probability that he travelled by train?

In a Kabaddi league, two matches are being played between Jaipur and Delhi. It is assumed that the outcomes of two games are independent. The probability of Jaipur winning, drawing and losing the game against Delhi are $\frac{1}{2}, \frac{3}{10}$, and $\frac{1}{5}$ respectively. Each team gets 5 points for win, 3 points for draw and 0 point for loss in a game. After two games, find the probability that:

A primary school teacher wants to teach the concept of 'larger number' to the students of Class II. To teach this concept, he conducts an activity in his class. He asks the children to select two numbers from a set of numbers given as 2, 3, 4, 5 one after the other without replacement. All the outcomes of this activity are tabulated in the form of ordered pairs given below:

	2	3	4	5
2		(2, 3)	(2, 4)	(2, 5)
3	(3, 2)		(3, 4)	(3, 5)
4	(4, 2)	(4, 3)		(4, 5)
5	(5, 2)	(5, 3)	(5, 4)

Evaluate: $P(A \cup B)$ if $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$

Teena is practising for an upcoming Rifle Shooting tournament. The probability of her shooting the target in the $1^{\text{st}}$, $2^{\text{nd}}$, $3^{\text{rd}}$ and $4^{\text{th}}$ shots are $0.4$, $0.3$, $0.2$ and $0.1$ respectively. Find the probability of at least one shot of Teena hitting the target.

A jewellery seller has precious gems in white and red colour which he has put in three boxes. The distribution of these gems is shown in the table given below:

Box	White	Red
I	1	2
II	2	3
III	3	1

He wants to gift two gems to his mother. So, he asks her to select one box at random and pick out any two gems one after the other without replacement from the selected box. The mother selects one white and one red gem. Calculate the probability that the gems drawn are from Box II.

A biased four-sided die with faces labelled 1, 2, 3 and 4 is rolled and recorded. Let $X$ be the result obtained when the die is rolled. The probability distribution for $X$ is given in the following table where $p$ and $q$ are constants.

$x$	1	2	3	4
$P(X=x)$	$p$	0·3	$q$	0·1

For the probability distribution, it is known that $E(X) = 2$. Find $p$ and $q$. Also, find $P(X > 2)$. Ajay plays a game with this four-sided die. In this game he is allowed a maximum of five rolls. His score is calculated by adding the results of each roll. He wins the game if his score is at least 10. After 3 rolls, Ajay has score of four points. Assuming that rolls of the die are independent, find the probability that Ajay wins the game.

The probability of the event $A$ occurring is $\frac{1}{3}$ and of the event $B$ occurring is $\frac{1}{2}$. If $A$ and $B$ are independent events, then find the probability of neither $A$ nor $B$ occurring.

A candidate takes three tests in succession and the probability of passing the first test is $\frac{1}{2}$. The probability of passing each succeeding test is $\frac{1}{2}$ or $\frac{1}{4}$ depending on whether he passes or fails in the preceding one. The candidate is selected, if he passes at least two tests. Find the probability that the candidate is selected.

$50$ tickets in a box are numbered $00, 01, 02, \dots, 49$. One ticket is drawn randomly from the box. Find the probability of the ticket having the product of its digits $7$, given that the sum of the digits is $8$?

A bag contains 19 tickets, numbered from 1 to 19. Two tickets are drawn randomly in succession with replacement. Find the probability that both the tickets drawn are even numbers.

A student answers a multiple choice question with $5$ alternatives, of which exactly one is correct. The probability that he knows the correct answer is $\frac{1}{5}$. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, find the probability that he did not tick the answer randomly.

A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.

A speaks truth in $60\\%$ of cases and B speaks truth in $90\\%$ of the cases. In what percentage of cases are they likely to contradict each other in stating the same fact?

A problem in Mathematics is given to three students A, B and C. Their chances of solving the problem are $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{4}$ respectively. Find the probability that

In a company, 15% of the employees are graduates and 85% of the employees are non-graduates. As per the annual report of the company, 80% of the graduate employees and 10% of the non-graduate employees are in the Administrative positions. Find the probability that an employee selected at random from those working in administrative positions will be a graduate.

If $A$ and $B$ are two events such that $P(A) > 0$ and $P(B) \neq 1$, then $P(\bar{A}/\bar{B})$ is

A bag contains 6 red and 5 blue balls and another bag contains 5 red and 8 blue balls. A ball is drawn from the first bag and without noticing its colour is placed in the second bag. If a ball is drawn from the second bag, then find the probability that the drawn ball is red in colour.