Question

8. We roll two fair dice. (1) Given that the roll results in a sum of 6 or less, what is the conditional probability that doubles are rolled? A "double" means that two dice have the same number (2) Given that the two dice land on different numbers, what is the conditional proba- bility that at least one die roll is a 1?

Answer 1

1)P(sum is 6 or less)=P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)=(1/36)+(2/36)+(3/36)+(4/36)+(5/36)=15/36

P(sum is 6 or less and double)=3/36 (as there are 3 events (1,1) ; (2,2) and (3,3) where sum is 6 or less and double)

hence P(double given sum is 6 or less)=P(sum is 6 or less and double)/P(sum is 6 or less)

=(3/36)/(15/36)=3/15=1/5

2)

P(rolls are different) =30/36 (as there are only 6 ways when roll are same)

P(rolls are different and one roll is 1) =10/36 (as 1 appears 11 times at least one and one time (1,1) it comes as double)

hence P( one roll is 1 given rolls are different)=P(rolls are different and one roll is 1)/P(rolls are different)

=(10/36)/(30/36)=10/30 =1/3