Let’s say the weatherwoman says the chance of a shower is 60% and the chance of a thunderstorm is 50%. What’s the chance of a shower OR a thunderstorm ? If you just add, you get:
P(shower or thunderstorm) = P(shower) + P(thunderstorm) = 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(shower and thunderstorm) = P(shower) × P(thunderstorm) = 30%
But we all know that showers and thunderstorms often occur together, so these two events are not independent either. Darn. (Insert stronger language here as desired).
In fact, we really can’t say ANYTHING about showers combined or not combined with thunderstorms. We can’t add the probabilities for shower OR thunderstorm, but we also can’t multiply to the probabilities for shower AND thunderstorm. 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!