A deck of cards includes 40 different cards. There are 8 cards in each of 5 suits. The cards are shuffled and a player receives 3 (different)
cards.
The probability that exactly 2 of these cards have the same suit is in
$(A) (0.46, 0.48].\\ (B) (0.44, 0.46].\\(C) (0.48, 0.50].\\(D) (A) (C) false.$
So, my approach is:
$P(\text{ 2 out of 3 are same suit })=3 P(\text{ picking card from any suit })P(\text{ picking card from the suit of the first card })P(\text{ pciking card of a different suit than the first 2 })=3(1)\left(\frac{7}{39}\right)\left(\frac{32}{38}\right)=3(0.151)=0.453$
So $B$ is correct?
Best Answer
Yes your answer is correct.
Here is another way to look at it. There are $5$ suits and there are exactly $2$ cards of the same suit. So we first choose the suit that the player gets two cards of. That is $5 \choose 1$. Now we choose $2$ cards from $8$ cards of that suit and the third card from remaining $32$ cards of other suits.
So the probability is,
$\displaystyle P = {5 \choose 1} {8 \choose 2} {32 \choose 1} \Big / {40 \choose 3} = \frac{7\cdot3\cdot32}{39\cdot38} = \frac{112}{247}$