Question

A balanced die is rolled twice. Are the events "the sum of the two rollsis 8" and "the first roll comes up 3" independent?

Answer 1

no they are not, but this will take some work

Answer 2

the probability that the sum is and 8 is \(\frac{5}{36}\) by counting the number of ways to get an 8 on two dice

Answer 3

the probability that the sum is an 8 given that the first die is a 3, here we also count.
the sample space now has 6 element namely
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)

of those 6, one has a total of 8 so the probability that you roll and 8 GIVEN that the first die is a 3 is \(\frac{1}{6}\)

Answer 4

now since \(\frac{5}{36}\neq \frac{1}{6}\) that means the events are DEPENDENT

Answer 5

i.e. not independent

Answer 6

events A, B are independent means \(P(A|B)=P(A)\) and since this is not the case here, they are dependent

Answer 7

Thanks a lot!!!!

Answer 8

yw