MathBench > Probability

Laws of AND and OR

What if you violate the Fine Print?

Let's say the weatherwoman says the chance of rain is 60% and the chance of snow is 50%. What's the chance of rain OR snow? If you just add, you get:

P(rain or snow) = P(rain) + P(snow) = 110%

Probabilities over 100% are meaningless. So right off the bat you know something is wrong.

How about the chance of rain AND snow? The Law of AND would say

P(rain and snow) = P(rain) * P(snow) = 30%

But we all know that rain often turns into snow and snow turns into rain, so these two events are not independent either. Darn. (Insert stronger language here as desired).

In fact, we really can't say ANYTHING about rain combined or not combined with snow. We can't add rain OR snow, but we also can't multiply to get rain AND snow. The two events aren't mutually exclusive, and they aren't independent. Neither law does us any good.

Again (as with the case of not being able to use OR on events that are not mutually exclusive), there are ways around this, if you have enough information. In this case what you would need to know is the conditional probability -- how much one event is conditioned on the other. However, we're not going to go into that, either!