Questions: A single card is drawn from a standard 52-card deck. Find the conditional probability that the card is a club, given that it is black. The probability that the card is a club, given that it is black is (Type an integer or a simplified fraction.)

Solution Steps

To find the conditional probability that the card is a club given that it is black, we need to use the definition of conditional probability. The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B). Here, A is the event that the card is a club, and B is the event that the card is black.

1. Calculate the probability of drawing a black card (P(B)).
2. Calculate the probability of drawing a club (P(A)).
3. Calculate the probability of drawing a black club (P(A ∩ B)).
4. Use the conditional probability formula to find P(A|B).

The probability of drawing a black card from a standard 52-card deck is given by:

\[ P(B) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = 0.5 \]

The probability of drawing a club from the same deck is:

\[ P(A) = \frac{\text{Number of clubs}}{\text{Total number of cards}} = \frac{13}{52} = 0.25 \]

Since all clubs are black, the probability of drawing a black club is:

\[ P(A \cap B) = \frac{\text{Number of clubs}}{\text{Total number of cards}} = \frac{13}{52} = 0.25 \]

Using the formula for conditional probability:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.25}{0.5} = 0.5 \]

Final Answer