Question

Use your common sense to answer this question. Are the events "Rain" and "Having a Car Accident" independent or dependent?

Answer 1

Dependent.

There are three equivalent definitions of independence. The two mathematical definitions are:

1) `P(A|B) = P(A) " for all "A" and "B`

2) `P(A and B) = P(A)*P(B)" for all "A" and "B`

However, the third definition is the most helpful for "common sense" questions like this:

3) Knowing the outcome of one variable gives you no information about the outcome of another.

Think about rain and car accidents. If I told you that it was raining heavily, would you think differently about my chances of being in a car accident than if I told you the sun was shining? If yes, then the two variables CANNOT be independent of each other and therefore must be dependent.