My question is about finite math

Question

Suppose E= "sum is 5" and F="exactly one of the dice is even". Are E and F independent? Explain why or why not.

Answer 1

Here's a general solution covering all possible cases (ie. rolling 2 to 5 dice)

Event E is really looking at the partitions of 5 (not including 5 itself as we must roll at least two dice) of which there are 6:

4+1

2+3

2+2+1

3+1+1

2+1+1+1

1+1+1+1+1

Now, for two events to be independent, the occurrence of one event should have no affect on the probability of the other event. With just this intuitive idea in mind, we can see from the partitions of 5 that if event E happens, then something can be said about event F. For example, in the case of rolling 5 dice, we certainly know F is not an impossible event (ie. P(F)>0), however, if E occurs, this can only happen if all five dice are 1, which means the probability of F given E is now 0! F cannot happen if E occurs. Similarly, in the case with three dice. In rolling two or four dice, if E occurs, then F must happen, whereas we know F is not a sure event on its own.