Question

A card is drawn from a deck of 52 playing cards. What is the probability that it is a king (4 king cards in a deck) or a red card (26 red cards in a deck)?

Reduce the fraction.

(1 point)
Responses

4134 over 13

3133 over 13

5135 over 13

713

Answer 1

To find the probability of drawing either a king or a red card from a standard deck of 52 playing cards, we can use the principle of inclusion-exclusion.

1. **Count the total number of kings**: There are 4 kings in the deck.

2. **Count the total number of red cards**: There are 26 red cards (13 hearts and 13 diamonds).

3. **Count the overlap**: The red cards contain 2 kings (the king of hearts and the king of diamonds). So, we must subtract these 2 from our total when using inclusion-exclusion.

Using the formula for inclusion-exclusion:
\[
P(K \cup R) = P(K) + P(R) - P(K \cap R)
\]

Where:
- \( P(K) = \frac{4}{52} \) (Probability of drawing a king)
- \( P(R) = \frac{26}{52} \) (Probability of drawing a red card)
- \( P(K \cap R) = \frac{2}{52} \) (Probability of drawing a red king)

Now substituting these values:
\[
P(K \cup R) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52}
\]
\[
= \frac{4 + 26 - 2}{52} = \frac{28}{52}
\]

Next, we can simplify this fraction:
\[
\frac{28}{52} = \frac{7}{13}
\]

Thus, the probability that the card drawn is a king or a red card is \( \frac{7}{13} \).

The correct response is:
**\( \frac{7}{13} \)**.