St.
Xavier’s High School
Test Paper
Topic : Probability
Time : 1.5 hours M. M. : 40
1. Let A and B be two given events such that P(A) = 0.6, P(B) = 0.2 and P(A/B) = 0.5. Then P(A′/B′) is
[1]
(a) 1/10 (b) 3/10 (c) 3/8 (d) 6/7
2. Let A and B be two given independent events such that P(A) = p and P(B) = q and P(exactly one of
A, B) = 2/3, then value of 3p + 3q – 6pq is: [1]
(a) 2 (b) -2 (c) 4 (d) –4
3. If P(A ∩ B) = 70% and P(B) = 85%, then P(A/B) is equal to [1]
(a) 14/17 (b) 17/20 (c) 7/8 (d) 1/8
4. Two dice are thrown once. If it is known that the sum of the numbers on the dice was less than 6
the probability of getting a sum 3 is: [1]
(a) 1/18 (b) 5/18 (c) 1/5 (d) 2/5
5. The probability distribution of the discrete variable X is given as: [1]
The value of k is:
(a) 8 (b) 16 (c) 32 (d) 48
6. The probability of A, B and C solving a problem are 1/2, 1/3 and 1/4 respectively. Then the
probability that the problem will be solved is [1]
(a) 1/2 (b) 3/4 (c) 1/4 (d) None
7. Three persons A, B and C, fire a target in turn. Their probabilities of hitting the target are 0.2, 0.3
and 0.5 respectively, the probability that target is hit, is [1]
(a) 0.993 (b) 0.94 (c) 0.72 (d) 0.90
8. Bag A contains 3 red and 5 black balls and bag B contains 2 red and 4 black balls. A ball is drawn
from one of the bags. The probability that ball drawn is red is: [1]
(a) 17/24 (b) 17/48 (c) 3/8 (d) 1/3
In the following questions 9 and 10, a statement of assertion (A) is followed by a statement of reason (R).
Mark the correct choice as:
(a) Both Assertion (A) and Reason (R) are true and Reason(R) is the correct explanation of assertion
(A).
(b) Both Assertion (A) and Reason (R) are true but Reason(R) is not the correct explanation of
assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
9. Assertion (A) : Three dice are rolled simultaneously. Consider the event E ‘getting a total of at
least 5’, F ‘getting the same number on all three dice’ and G ‘getting a total of 15’. Then E, F and G
are mutually independent events. [1]
Reason (R) : Three events A, B and C are said to be mutually independent, if
P(A ∩ B) = P(A) P(B)
P(A ∩ C) = P(A) P(C)
P(B ∩ C) = P(B) P(C)
and P(A ∩ B ∩ C) = P(A) P(B) P(C)
If at least one of the above conditions is not true for the given events, then events are not
independent.
10. Assertion: If A and B are two mutually exclusive events with P(A’) = 5/6 and P(B) = 1/3. Then
P(A|B ) is equal to 1/4. [1]
Reason (R) : If A and B are two events such that P(A) = 0.2, P(B) = 0.6 and P(A|B) = 0.2 then the
value of P(A|B’) is 0.2.
11. Events A and B are such that P(A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4. State whether A
and B are independent? [2]
12. The probability of simultaneous occurrence of at least one of the two events A and B is p. If the
probability that exactly one of A, B occurs is q, then prove that P(A′) + P(B′) = 2 – 2p + q. [2]
13. If P(A) = 3/5, P(B) = 1/5, find (A∩B), if [2]
(i) A and B are mutually exclusive
(ii) A and B are independent
14. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8,
given that the red die resulted in a number less than 4. [2]
15. A bag contains (2n + 1) coins, out of which n coins have head on both the sides and rest are fair
coins. A coin is selected at random and is tossed, if it results in a head with probability 31/42, find
the value of n. [3]
16. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what
is the conditional probability that both are girls given that
(i) the youngest is a girl? (ii) atleast one is a girl? [3]
17. The random variable X has a probability distribution P(X) of the following form, where k is some
number: [3]
k , if x 0
2k , if x 0
PX .
3k , if x 0
0, otherwise
(a) Determine the value of k.
(b) Find P(X < 2), P(X ≤ 2), P(X ≥ 2).
OR
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find
the probability distribution of number of tails.
18. A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards
are drawn at random (without replacement) and are found to be all spades. Find the probability of
the lost card being a spade. [5]
OR
Two numbers are selected at random (without replacement) from the first six positive integers.
Let X denote the larger of the two numbers obtained. Find the probability distribution of the
random variable X, and hence find the mean of the distribution.
19. Case-Study 1:
In a test, you either guesses or copies or knows the answer to a multiple – choice question with
four choices. The probability that you make a guess is 1/3, you copy the answer is 1/6. The
probability that your answer is correct, given that you guess it, is 1/8. And also, the probability
that you answer is correct, given that you copy it, is 1/4.
(i) The probability that you know the answer. [1]
(ii) Find the probability that your answer is correct given that you guess it and the probability
that your answer is correct given that you know the answer. [1]
(iii) Find the probability that you know the answer given that you correctly answered it. [2]
OR
(iii) Find the total probability of correctly answered the question. [2]
20. Case-Study 2:
Read the following passage and answer the questions given below.
Final exams are approaching, so Mr. Kumar decided to check the preparation of the few weak
students in the class. He chooses four students A, B, C and D then a problem in mathematics is
given to those four students A, B, C, D. Their chances of solving the problem, respectively, are 1/3,
1/4, 1/5 and 2/3.
Based on the given information answer the following questions. What is the probability that:
(i) the problem will be solved? (2)
(ii) at most one of them solve the problem? (2)
