Question

two cards are drawn without replacement from a standard deck of 52 cards. what is the probability that the second card is a heart, given that the first card is a heart?

Explanation:

Step1: Analyze the remaining cards

After drawing one heart (first card), there are \(52 - 1=51\) cards left in the deck.

Step2: Analyze the remaining hearts

Originally, there are 13 hearts. After drawing one heart, there are \(13 - 1 = 12\) hearts left.

Step3: Calculate the probability

The probability that the second card is a heart given the first is a heart is the number of remaining hearts divided by the number of remaining cards. Using the formula for conditional probability \(P(A|B)=\frac{n(A\cap B)}{n(B)}\), here \(n(A\cap B)\) (number of ways first is heart and second is heart) and \(n(B)\) (number of ways first is heart). The probability \(P=\frac{12}{51}\).

Answer:

\(\frac{12}{51}\)