Curious die, stats

Question

A peculiar die has the following properties: on any roll the probability of rolling either a 2, a 1, or a 3 is 1/2, just as it is with an ordinary die. Moreover, the probability of rolling either a 3, a 4, or a 5 is again 1/2. However, the probability of rolling a 3is 7/16, not 1/6 as one would expect of an ordinary fair die.

From what you know about this peculiar die, compute the following.

1. The probability of rolling either a 6, a 4, or a 5
2. The probability of rolling neither a 2 nor a 1

Answer 1

Prob(3) = 7/16 is given. Assuming the other probabilities are equally likely:

Prob(1,2,or 3) = 1/2 = 8/16 ---> prob(1 or 2) = 8/16-7/16 = 1/16, since prob(3) = 7/16

Prob(1,2,or 5) = 1/2 = 8/16 ---> prob(5) = 8/16 - 1/16 = 7/16

Assuming they are equally likely, Prob(1) = 1/32

Prob(2) = 1/32

Prob(3) = 7/16 = 14/32

Prob(4) = 1/32

Prob(5) = 7/16 = 14/32

Prob(6) = 1/32

Prob(4,5or 6) = 16/32 = 1/2 while prob(3,4,5,6) = 30/32 = 15/16