Question

a deck of cards has 4 suits: hearts, clubs, diamonds, and spades. each suit has 13 cards numbered from 1 to 13. a game is played in which a card is drawn from the deck and played face - up, then each player takes turns drawing additional cards from the deck. the first player to draw a card matching the suit or number of the face - up card wins. cards are not returned to the deck after being drawn. if the first player does not win on their first turn, what is the probability that the second player will win on their first turn?
a. $\frac{3}{13}$
b. $\frac{15}{51}$
c. $\frac{3}{10}$

Explanation:

Step1: Calculate total number of cards

A standard deck has \(52\) cards.

Step2: Calculate number of favorable cards

Let the face - up card be of a particular suit and number.
Number of cards with the same suit: \(12\) (since one card of that suit is already face - up).
Number of cards with the same number: \(3\) (one for each of the other three suits).
But we have double - counted the card which has both the same suit and number (there is no such card in this case as the first card is not in the deck for the second draw). So the number of favorable cards \(n=12 + 3=15\).

Step3: Calculate probability

The number of remaining cards \(N = 51\).
The probability \(P=\frac{15}{51}\).

Answer:

B. \(\frac{15}{51}\)