Questions: We are drawing a single card from a standard 52-card deck. Find the following probability. P (club black) The probability is . (Type an integer or a simplified fraction.)

Solution Steps

To find the probability of drawing a club given that the card is black, we need to use conditional probability. A standard deck has 26 black cards (13 clubs and 13 spades). The probability of drawing a club given that the card is black is the number of club cards divided by the total number of black cards.

In a standard 52-card deck, there are 26 black cards, which consist of 13 clubs and 13 spades. Thus, we have: \[ \text{Total black cards} = 26 \]

Among the black cards, there are 13 clubs. Therefore: \[ \text{Number of club cards} = 13 \]

The probability of drawing a club given that the card is black can be calculated using the formula for conditional probability: \[ P(\text{club} | \text{black}) = \frac{P(\text{club and black})}{P(\text{black})} \] Since all clubs are black, we have: \[ P(\text{club and black}) = \frac{13}{52} \quad \text{and} \quad P(\text{black}) = \frac{26}{52} \] Thus, the conditional probability simplifies to: \[ P(\text{club} | \text{black}) = \frac{13/52}{26/52} = \frac{13}{26} = \frac{1}{2} \]

Final Answer