Question

two dice are tossed; what is the probability that the numbers are the same on both dice if it is known that the sum of the spots is 6

Answer 1

When two dice are tossed, then we get the sample space,

S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1,(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6))}. So, n(S)=36

Let us define an event A as numbers are the same on both dice i.e., A={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}. So, n(A)=6

Let us define an event B as sum of the spots is 6 i.e. B={(1,5),(2,4),(3,3),(4,2),(5,1)}. So, n(B)=5

Thus, we have A⋂B={(3,3)}

P(A)=6/36=1/6, P(B)=5/36 and P(A⋂B)=1/36

We have to find the probability that the numbers are the same on both dice if it is known that the sum of the spots is 6 i.e. P(A|B)

By conditional probability, we can write,

P(A|B)=(P(A⋂B))/P(B) = (1/36)/(5/36) =1/5