Question

Two fair dice are rolled once in such a way that the probability of summation of the outcome on both of them having a prime number is three times than the other case. Calculate the probability of getting any prime number as the summation on the faces of both the dice.

Answer 1

Let (X,Y) denote the outcome on the pair of dice

Total number of possible outcomes = 36

Cases where summation is prime (i.e. x + y is prime or 2,3,5,7,11) are {(1,1), (2,1), (1,2), (1,4), (2,3), (3,2), (4,1), (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (5,6), (6,5)}

15 total cases

Let the probability of non prime summation outcomes be p

Thus, the probability of prime summation outcomes = 3p

Thus, 15*3p + (36 - 15)*p = 1

-> p = 1/66

The probability of getting any prime number as the summation on the faces of both the dice = 15*3p

= 45/66 = 15/22 = 0.6818