Commenters Sylvain B., xpil, and Christian Luca all gave correct answers to the challenge from my previous post. If the probability that someone likes X is , then the probability they don’t like X is . Therefore the probability that someone doesn’t like anchovies and doesn’t like books is . (If liking anchovies and books are independent, then not liking those things is also independent.) Likewise, the probability that someone doesn’t like all three is .
The reason I asked the question, however, is to see the pattern that emerges if we multiply out these expressions, which commenter Buddha Buck explained.
The first line is the result of multiplying out ; the second is ; and the last is of course . After expanding everything as much as possible, we get one term for the product of every possible combination of the probabilities, with a positive sign for an even number of probabilities, and negative for an odd number.
Do you see why this happens? Each term in the result arises from choosing one of the two possible terms from each parenthesized expression. Taking as a specific example, we get if we choose from , and from , and from , yielding .
And now for a set of follow-up questions. Suppose that is the set of all people in our population, is the set of people who like anchovies, and is the set of people who like books. Suppose we also know how many people like both anchovies and books, that is, we know . (The notation denotes the size of a set , and denotes the intersection of two sets, that is, the set of elements the two sets have in common.)
How many people like neither anchovies nor books?
Now suppose is the set of people who like carpets. Suppose we also know the sizes of the sets , , and . How many people like none of the things?
And so on?