Let A and B be the events such that

Let A and B be the events such that

and

Find

(i) P(A ∩ B)

(ii) P(A / B)

(iii) P(B / A)

(iv)

Let A and B be the events such that

and

Find

(i) P(A ∩ B)

(ii) P(A ∪ B)

(iii) P(A / B)

Let A and B be the events such that

and

Find

(i) P(A ∩ B)

(ii) P(A ∪ B).

A die is rolled. If the outcome is an even number, what is the probability that it is a number greater than 2?

A coin is tossed twice. If the outcome is at most one tail, what is the probability that both head and tail have appeared?

Three coins are tossed simultaneously. Find the probability that all coins show heads if at least one of the coins shows a head.

Two unbiased dice are thrown. Find the probability that the sum of the numbers appearing is 8 or greater, if 4 appears on the first die.

A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?

Two dice were thrown and it is known that the numbers which come up were different. Find the probability that the sum of the two numbers was 5.

A coin is tossed and then a die is thrown. Find the probability of obtaining a 6, given that a head came up.

A couple has 2 children. Find the probability that both are boys if it is known that (i) one of the children is a boy, and (ii) the elder child is a boy.

In a class, 40% students study mathematics; 25% study biology and 15% study both mathematics and biology. One student is selected at random. Find the probability that

(i) he studies mathematics if it is known that he studies biology

(ii) he studies biology if it is known that he studies mathematics.

The probability that a student selected at random from a class will pass in Hindi is and the probability that he passes in Hindi and English is What is the probability that he will pass in English if it is known that he has passed in Hindi?

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a coat is 0.3 and the probability that he will buy a shirt given that he buys a coat is 0.4. Find the probability that he will buy both a shirt and a coat.

In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.

(i) Find the probability that he reads neither Hindi nor English news paper.

(ii) If he reads Hindi newspaper, what is the probability that he reads English newspaper?

(iii) If he reads English newspaper, what is the probability that he reads Hindi newspaper?

Two integers are selected at random from integers 1 through 11. If the sum is even, find the probability that both the numbers selected are odd.

A bag contains 17 tickets, numbered from 1 to 17. A ticket is drawn, and then another ticket is drawn without replacing the first one. Find the probability that both the tickets may show even numbers.

Two marbles are drawn successively from a box containing 3 black and 4 white marbles. Find the probability that both the marbles are black if the first marble is not replaced before the second draw.

A card is drawn from a well-shuffled deck of 52 cards and without replacing this card, a second card is drawn. Find the probability that the first card is a club and the second card is a spade.

There is a box containing 30 bulbs, of which 5 are defective. If two bulbs are chosen at random from the box in succession without replacing the first, what is the probability that both the bulbs are chosen are defective?

A bag contains 1.0 white and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that the first ball is white and the second is black?

An urn contains 5 white and 8 black balls. Two successive drawings of 3 balls at a time are made such that the balls drawn in the first draw are not replaced before the second draw. Find the probability that the first draw gives 3 white balls and the second draw gives 3 black balls.

If E₁ and E₂ are the two events such that and show that E₁ and E₂ are independent events.

If E₁ and E₂ are independent events such that P(E₁) = 0.3 and P(E₂) = 0.4, find

(i) P(E₁∩ E₂)

(ii) P(E₁∩ E₂)

(iii)

(iv)

Let A and B be the events such that and P(not A or not B)

State whether A and B are

(i) mutually exclusive

(ii) independent

Kamal and Vimal appeared for an interview for two vacancies. The probability of Kamal’s selection is 1/3, and that of Vimal’s selection is 3. Find the probability that only one of them will be selected.

Arun and Ved appeared for an interview for two vacancies. The probability of Arun’s selection is 1/4, and that of Ved’s rejection is 2/3. Find the probability that at least one of them will be selected.

A and B appear for an interview for two vacancies in the same post. The probability of A’s selection is 1/6 and that of B’s selection is 1/4. Find the probability that

(i) both of them are selected

(ii) only one of them is selected

(iii) none is selected

(iv) at least one of them is selected.

Given the probability that A can solve a problem is 2/3, and the probability that B can solve the same problem is %, find the probability that

(i)at least one of A and B will solve the problem

(ii)none of the two will solve the problem

A problem is given to three students whose chances of solving it are 1/4, 1/5 and 1/6, respectively. Find the probability that the problem is solved.

The probabilities of A, B, C solving a problem are 1/3, 1/4 and 1/6, respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them will solve it.

A can hit a target 4 times in 5 shots, B can hit 3 times in 4 shots, and C can hit 2 times in 3 shots. Calculate the probability that

(i) A, B and C all hit the target

(ii) B and C hit and A does not hit the target.

Neelam has offered physics, chemistry and mathematics in Class XII. She estimates that her probabilities of receiving a grade A in these courses are 0.2, 0.3 and 0.9 respectively. Find the probabilities that Neelam receives

(i) all A grades

(ii) no A grade

(iii) exactly 2 A grades.

An article manufactured by a company consists of two parts X and Y. In the process of manufacture of part X. 8 out of 100 parts may be defective. Similarly, 5 out of 100 parts of Y may be defective. Calculate the probability that the assembled product will not be defective.

A town has two fire-extinguishing engines, functioning independently. The probability of availability of each engine when needed is 0.95. What is the probability that

(i) neither of them is available when needed?

(ii) an engine is available when needed?

A machine operates only when all of its three components function. The probabilities of the failures of the first, second and third components are 0.14, 0.10 and 0.05, respectively. What is the probability that the machine will fail?

An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shots are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that at least one shot hits the plane?

Let S₁ and S₂ be the two switches and let their probabilities of working be given by P(S₁) = 4/5 and P(S₂) = 9/10. Find the probability that the current flows from the terminal A to terminal B when S₁ and S₂ are installed in series, shown as follows:

Let S₁ and S₂ be two the switches and let their probabilities of working be given by P(S₁) = 2/3 and P(S₂) = 3/₄. Find the probability that the current flows from terminal A to terminal B, when S₁ and S₂ are installed in parallel, as shown below:

A coin is tossed. If a head comes up, a die is thrown, but if a tail comes up, the coin is tossed again. Find the probability of obtaining

(i) two tails

(ii) a head and the number 6

(iii) a head and an even number.