Question

5. Suppose the six-sided die you are using for this problem is not fair. It is biased so that rolling a 6 is three times more likely than any other roll. For this problem, the experiment is rolling a six-sided die twice.

(A): What is the probability that one or both rolls are even numbers (2, 4 or 6’s)?

(B): What is the probability that at least one of the rolls is an even number or that the sum of the two rolls is less than 6?

Thank you in advance :)

Answer 1

Let the probability of rolling 1, 2, 3, 4 and 5 be p

Then, probability of rolling 6 is 3p

Since sum of probabilities = 1

5p + 3p = 1

p = 1/8 = 0.125

P(1) + P(2) + P(3) + P(4) + P(5) = 0.125

P(6) = 3x0.125 = 0.375

(A) P(odd number) = P(1) + P(3) + P(5)

= 3 x 0.125

= 0.375

P(one or both rolls are even numbers) = 1 - P(both are odd numbers)

= 1 - 0.375x0.375

= 0.8594

(B) P(at least one of the rolls are even or the sum is less than 6)

= P(one or both rolls are even) + P(both rolls are odd and sum is less than 6)

= 0.8594 + P[(1,1), (1,3), (3,1)]

= 0.8594 + 3x(0.125x0.125)

= 0.9063