A pair of fair dice is rolled. Let E denote the event that the number landing uppermost on the first die is a 3, and let F denote the event that the sum of the numbers landing uppermost is 6. Determine whether E and F are independent events

Answer 1

Of course they are not independent. The ways to get six total on both die are
5,1
1,5
4,2
2,4
3,3

or five ways. If the first three is secluded, there are only four ways to get six.

Pr(E)=1/6
Pr(F)=5/36
Pr(F|E)= 1/6

Two events, E and F, are independent if the fact that E occurs does not affect the probability of F occurring. Here, The probability of F depends on whether or not E has occurred.