# Determining whether two events are Independent

Two fair dice are rolled, and events A and B are as follows: Event A occurs if the largest number showing is at most 3. Event B occurs if neither 1 nor a 6 is showing Are they Independent?

I said that they are dependent because if Event A occurs where let's say a 1 occurs, then Event B does not occur. However, what happens when 2 occurs because then Event A occurs and Event B occurs as well.

• Intuition is often reliable when it comes to dependence/independence. But not always, so I would advise calculating. Jun 16 '14 at 20:28
• As @andrenicolas has suggest compute the probabilities, specifically the event space and see if the sets are independent I.e what values of the dies are permitted in the space for events A and then look at B. Jun 16 '14 at 21:35

## 1 Answer

$A$ and $B$ are independent iff $$\Pr[A \cap B]=\Pr[A] \times \Pr[B],$$ where $\Pr[A \cap B]$ is the probability that both $A$ and $B$ occur simultaneously.

So, given this information, you tell me. Are $A$ and $B$ independent?

• No they are not because the Prob(AnB) is 1/6 and the prob(A)*Prob(B) is .22222 Jun 16 '14 at 20:38