Question

7. (3 points) Given a fair 6-sided die. Each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 6 are all equal. 1) If it is rolled once and let A be the event of rolling a number larger than 3 and B be the event of rolling an odd number. What is P(AV B)? 2) If it is rolled three times, what is the probability that the same number shows up in all three times? 3) If it is rolled twice, what is the probability that the sum of the two numbers is an even number?

Answer 1

Given a fair 6-sided die, each time the die is rolled, probabilities of rolling any of the number from 1 to 6 are equal.

1) If it is rolled once, A = event of rolling a number > 3, B = event of rolling an odd number.

A={4,5,6}, B={1,3,5}, A and B = {5}

P(A)= 3/6=1/2, P(B)= 3/6=1/2, P(A and B)= 1/6

$P(A \cup B)= P(A)+P(B)-P(A \cap B)= 1/2+1/2 - 1/6 = 5/6$

2) If it is rolled thrice, probability that the same number shows three times = 6/ 6³ = 1/36

This is because, the total number of possible outcomes = $6^3$

and the even that the same number will occur in all three can only be as = (x,x,x) for x=1,2,3,4,5,6. Hence, 6 favourable outcomes.

3) If the die is rolled twice, total number of possible outcomes = $6^2$

outcomes where sum of the two numbers is an even number = {(1,1),(1,3),(1,5),(2,2).(2,4),(2,6).(3,1),(3,3),(3,5),(4,2),(4,4),(4,6),(5,1),(5,3),(5,5),(6,2),(6,4),(6,6)} i.e. 18 favourable outcomes

probability that sum of the two numbers is an even number = 18/36 = 1/2