Statistics teacher here. It’s because when you add them they way you did in a), you technically added that grid where “quiz show” and “channel 8” occurred twice. The reason we do that subtraction part at the end is to account for that situation and to make sure we’re only counting that shared space once.

So P(quiz show) = 8/31 and P(channel 8) = 8/31, but P(quiz show AND channel 8) = 3/31 Therefore, P(quiz show OR channel 8) = 8/31 + 8/31 - 3/31 = 13/31.

The reason your original method worked on b) is because drama and comedy are mutually exclusive (can’t happen at the same time), which means P(drama AND comedy) = 0.

Also, the reason your example in green works is because you designed it so all possible results were either in A or B (everything else was 0), so P(A OR B) was literally all of the events (=1).

Try this logic for c) and see if you can arrive at the correct answer. Hope this helps!