Probability — Practice

Question 1

A die is thrown once. Find the probability of getting a number greater than 4.

Accepted Answer

S={1,2,3,4,5,6}, n(S)=6. Favourable outcomes {5,6}, n(E)=2. Answer: P(E) = 2/6 = 1/3.

Question 2

A die is thrown. Let A = 'an odd number appears' and B = 'a number greater than 4 appears'. Are A and B mutually exclusive? Are they exhaustive?

Accepted Answer

A={1,3,5}, B={5,6}. A intersection B={5}, which is not empty, so they are not mutually exclusive. A union B={1,3,5,6}, which is not all of S={1,2,3,4,5,6} (missing 2,4), so they are not exhaustive either. Answer: A and B are neither mutually exclusive nor exhaustive.

Question 3

A card is drawn at random from a standard deck of 52 cards. Find the probability that it is a King or a Heart.

Accepted Answer

P(King)=4/52, P(Heart)=13/52, P(King and Heart)=P(King of Hearts)=1/52. Answer: P(King or Heart) = 4/52+13/52-1/52 = 16/52 = 4/13.

Question 4

Two dice are thrown. Find the probability that the sum is at least 4, using the complement.

Accepted Answer

It's easier to find P(sum<4) = P(sum=2 or 3). Sum=2: {(1,1)}, 1 way. Sum=3: {(1,2),(2,1)}, 2 ways. Total 3 ways out of 36. So P(sum<4)=3/36=1/12. Answer: P(sum>=4) = 1-1/12 = 11/12.

Question 5

A sample space is S={w1,w2,w3,w4}. Check whether the assignment P(w1)=0.4, P(w2)=0.3, P(w3)=0.2, P(w4)=0.05 is a valid probability assignment.

Accepted Answer

Each value is between 0 and 1. Sum = 0.4+0.3+0.2+0.05 = 0.95, which is not equal to 1. Answer: This is not a valid probability assignment, since the probabilities do not sum to 1.

Question 6

Two dice are thrown simultaneously. Find the probability that the sum is at least 9.

Accepted Answer

n(S)=36. Favourable: sum=9 has 4 ways {(3,6),(4,5),(5,4),(6,3)}; sum=10 has 3 ways; sum=11 has 2 ways; sum=12 has 1 way. Total favourable = 4+3+2+1=10. Answer: P(sum>=9) = 10/36 = 5/18.

Question 7

In a group of students, 50% study Physics, 40% study Chemistry, and 20% study both. If a student is picked at random, find the probability that the student studies Physics or Chemistry.

Accepted Answer

P(Physics)=0.5, P(Chemistry)=0.4, P(both)=0.2. Answer: P(Physics or Chemistry) = 0.5+0.4-0.2 = 0.7.

Question 8

A coin is tossed three times. Give an example of two events that are mutually exclusive but not exhaustive, and justify both properties.

Accepted Answer

Let A = 'exactly two heads' = {HHT,HTH,THH} and B = 'exactly two tails' = {HTT,THT,TTH}. A intersection B is empty (no outcome has exactly two heads and exactly two tails at once), so they are mutually exclusive. Answer: A union B = {HHT,HTH,THH,HTT,THT,TTH}, which excludes HHH and TTT, so A and B are not exhaustive.

Practice — Probability

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