We add because if we simply subtract |A| and |B| from |P|, we end up subtracting the intersection twice, and so need to counteract this to obtain an accurate value.

Similarly, if we wish to know how many people like neither anchovies nor books nor carpet, this is the size of the intersection of the complements of A, B, and C; that is, . Again applying De Morgan’s Law, this is equal to , which, assuming we also know |C|, is equal to

Here we add the sizes of the pairwise intersections of sets for the same reason as in the previous paragraph, but in doing so overshoot on ; the size of this subset has been subtracted three times from |P| and then added back three times with , , and . Thus we need to subtract one last time to obtain an accurate value.

In general, let P be the set of all people in our population and let be a finite collection of subsets of P containing people who like thing i. Then, following the pattern and assuming we know the sizes of P and of every finite intersection of sets , the number of people who like none of the things should be equal to

This is in direct correlation with the result outlined in the blog post (where ), suggesting a relationship between multiplication of probabilities and intersection of sets.

]]>The number of people who like neither anchovies, books, nor carpets would be P – (|A $\cap C| + |B $\cap C| + |A $\cap B $\cap C|).

The number of people who do not like any of the enumerated things considered, which we can call n could be denoted by: P – (|A_1 $\cap A_2| + |A_2 $\cap A_3| + |A_3 $\cap A_4| + …|A_n-1 $\cap A_n| + |A_1 $\cap A_2 $\cap A_3 $\cap A_4…$\cap A_n). |

]]>The probability of a random person not liking any of the three aforementioned things would be: (1-a)*(1-b)*(1-c).

The generalization for four, five or more independent probabilities for a person not liking any of the things would be (1-a)*(1-b)*(1-c)*(1-d) for four things, (1-a)*(1-b)*(1-c)*(1-d)*(1-e) for five things, and for more n things would be (1-a_1)*(1-a_2)*…(1-a_n).

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