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

Jan 1, 2015

Dependent.

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

1) $P \left(A | B\right) = P \left(A\right) \text{ for all "A" and } B$

2) $P \left(A \mathmr{and} B\right) = P \left(A\right) \cdot P \left(B\right) \text{ for all "A" and } B$

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.