A homework questions asks:
John has two quarters and two dimes. He tosses these four coins 3 times. Let U be the number of heads of quarters, and V be the number of heads of dimes. Denote X = U + V and Y = |U − V |. Find P(X = 3, Y = 1).
The given answer is: .25
I'm not sure how to get there. I figured if X = U + V, and we want X = 3, then the possible pairings are: (0,3) (1,2) (2,1) (3,0)
If Y = |U-V| and we want Y = 1, then the possible pairings are: (0,1) (1,2) (2,3) (3,4) (4,5) (5,6) (6,5) (5,4) (4,3) (3,2) (2,1) (1,0)
But after getting all the pairings I'm blanking on how to get the probability. I feel like it's probably really simple too, but I just can't seem to remember how to get to the final answer.
Any help?
There are only 2 ways for X=3 and Y=1, namely (1,2) and (2,1). Each of these has 2 ways of happening (2 heads for quarters and either head for dimes and vice versa). There are$2^4=16$ possible outcomes all together. Probability therefore is 1/4.
Tossing 3 times doesn't effect probability. If X and Y are the sum for all 3 sets of tosses, then it is a different question.