Rainy Days

One of the most common errors people make when thinking about probability is to add up probabilities that seem independent in order to find the combined probability.

If there is a 20% chance of rain tomorrow, and a 20% chance of rain the next day, what is the probability that in will rain at least once in the next two days?

One thinks that 20% each day implies a 40% chance in two days. (This is the classic P = np error.) This of course implies that in five days, the chance of rain is 100%, and in six days ...


Now let us consider the matter a trifle more more rigorously.

Always remember to check the behavior of your expressions for the following case.