I suck at probability. I can't think of ways of understanding any problems that involve probability. Here's the problem I'm currently stuck on:

So the first thing I did was make it a little easier for me to understand by rewriting the table in terms of frequency:

Code (Text):

WING SIZE
EYE COLOR Normal Miniature
Normal 4/15 1/50
Vermillion 1/100 151/300

Now I've thought of a method for going about solving the problem but it doesn't work. Isn't it correct that P(A or B) = P(A) + P(B)??? So I figured that the answer would be P(VermillionEyes or MiniatureWings or Both) = P(VermillionEyes) + P(MiniatureWings) + P(Both). That would be 154/300 + 157/300 + 151/300. BUT THIS IS OBVIOUSLY WRONG SINCE IT GOES PAST 300/300!

How do I do this? Why is the P(A or B) = P(A) + P(B) thing not working?

A total of 3+ 151= 154 flies have vermilion eyes (now that I would like to see!) and another 6 flies have minature wings so 154+ 6= 160 flies have "either vermillion eyes or miniature wings, out of 300. The probability of that is 160/300= 16/30= 8/15.

Another way to do that is to say the in order not to have "vermilion eyes or miniature wings or both" a fly must have both normal wings and normal eyes. There are 140 such flies so the probability of that is 140/300= 14/30= 7/15. The probability that that is not true is 1- 7/15= (15-7)/15= 8/15 again.

Your mistake is in saying "(VermillionEyes or MiniatureWings or Both) = P(VermillionEyes) + P(MiniatureWings) + P(Both)"

The correct formula is (VermillionEyes or MiniatureWings or Both) = P(VermillionEyes) + P(MiniatureWings) - P(Both). Do you see the minus sign? The reason is that "vermilion eyes" includes "minature wings" and "miniature wings" includes "vermilion eyes". P(Vermilion eyes) and P(MinatureWings) includes both twice. You need to subtract one of them, not add it in again!