Calculus Multiplicative Principles

Question

A group of college students were asked about their TV watching habits. Of those surveyed, 34 watch The Walking Dead, 38 watch The Blacklist, and 36 watch Game of Thrones. Additionally, 7 watch The Walking Dead and The Blacklist, 11 watch The Walking Dead and Game of Thrones, and 12 watch The Blacklist and Game of Thrones. There are 4 students who watch all three shows.

How many students surveyed watched at least one of the shows?

Answer 1

If you make a Venn Diagram and subtract out the various pieces, you'll end up with something like this:

Notice that the overlap between all three shows is 4. The overlap between The Walking Dead and Game of Thrones is 11 (7 + 4), the overlap between The Walking Dead and The Blacklist is 7 (3 + 4), and the overlap between The Blacklist and Game of Thrones is 12 (4 + 8).

Considering those within The Walking Dead circle, there are 14 who also watch other shows (7 + 4 + 3) meaning there must be 20 that ONLY watch The Walking Dead. (20 + 14 = 34 total).

Considering those within The Blacklist circle, there are 15 who also watch other shows (8 + 4 + 3) meaning there must be 23 that ONLY watch The Blacklist. (23 + 15 = 38 total).

Considering those within Game of Thrones circle, there are 19 who also watch other shows (7 + 4 + 8) meaning there must be 17 that ONLY watch Game of Thrones. (19 + 17 = 36 total).

So to determine how many watch at least one show, just count up all the individual pieces:

20 + 17 + 23 + 7 + 3 + 8 + 4 = 82 people who at least watch one of the shows.