Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a heart given that the second card is a diamond?

Question

zarkon · Answer

$$P(H_1|D_2)=\frac{P(H_1,D_2)}{P(D_2)}=\frac{P(D_2|H_1)P(H_1)}{P(D_2)}$$

anonymous · Answer

.25?

zarkon · Answer

that is not what i get

zarkon · Answer

$$=\frac{P(D_2|H_1)P(H_1)}{P(D_2)}$$
$$=\frac{P(D_2|H_1)P(H_1)}{P((D_2,D_1)\cup(D_2,D_1'))}$$
$$=\frac{P(D_2|H_1)P(H_1)}{P(D_2,D_1)+P(D_2,D_1')}$$
$$=\frac{P(D_2|H_1)P(H_1)}{P(D_2|D_1)P(D_1)+P(D_2|D_1')P(D_1')}$$

zarkon · Answer

$$=\frac{\frac{13}{51}\frac{13}{52}}{\frac{12}{51}\frac{13}{52}+\frac{13}{51}\frac{39}{52}}=\frac{13}{51}$$

anonymous · Answer

H1 would not be 1/4?

zarkon · Answer

13/52=1/4

anonymous · Answer

duh. sorry I am trying to figure out my mistake

anonymous · Answer

you might notice that the probability that first card is a heart given that the second card is a diamond is exactly the same as the probability that the second card is a heart given the first card is a diamond

amistre64 · Answer

knowing one card is a diamond; doesnt that mean there are 13 hearts out of 51 left?

anonymous · Answer

Zarkon, Hwo do you come up with 39/52?

zarkon · Answer

the are 13 diamonds..so there are 52-13=39 that are not diamonds...out of 52 cards

anonymous · Answer

got it. Thanks