Probability of dependent event- marble problem

Question

Question : there are 4 blue marble labelled P, 7 blue marbles labelled Q, 3 green marbles labelled P, 9 green marbles labelled Q. Two marbles are drawn at random from the box. Find the probability that both the marbles are of the same colour but labelled iwth a different letter.

 

Why is the answer 4/23 x 7/22 + 3/23 x 9/22 = 5/46 ?

 

I think the answer should be

Porbability of (first ball blueP & second ball blueQ, or first ball blueQ & second ball blueP, or first ball greenP & seocnd ball greenQ , or first ball greenQ & second ball greenP ) = 4/23 x 7/22  + 7/23 x 4/22 + 3/23 x 9/22 + 9/23 x 3/22 = 5/23

Why is my answer wrong?

Answer 1

You don't account for all 4 possibilities that way. As noted, you have a 4/23 shot of picking a blue P first, then a 7/22 shot of picking a blue Q. One happens or the other and not both, so...

 

4/23 x 7/22 or 7/23 x 4/22, etc.