Question

A standard deck of 52 cards contains four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. Two cards are randomly drawn without replacement. Calculate the probability of drawing two diamond cards. Express your answer in percent form rounding to the nearest hundredth.(1 point)
%

Answer 1

To calculate the probability of drawing two diamond cards from a standard deck of 52 cards without replacement, we follow these steps:

1. **Determine the total number of diamond cards**: There are 13 diamond cards in a deck.

2. **Calculate the probability of drawing the first diamond card**:
\[
P(\text{first diamond}) = \frac{13}{52}
\]

3. **After drawing the first diamond card, determine the number of remaining diamond cards and total cards**:
- Remaining diamond cards = 12
- Remaining total cards = 51

4. **Calculate the probability of drawing the second diamond card**:
\[
P(\text{second diamond | first diamond}) = \frac{12}{51}
\]

5. **Calculate the joint probability of both events** (drawing two diamond cards):
\[
P(\text{two diamonds}) = P(\text{first diamond}) \times P(\text{second diamond | first diamond}) = \frac{13}{52} \times \frac{12}{51}
\]
\[
P(\text{two diamonds}) = \frac{13 \times 12}{52 \times 51} = \frac{156}{2652}
\]

6. **Simplify the fraction**:
\[
\frac{156}{2652} = \frac{1}{17} \text{ (This can be confirmed since 156 and 2652 both divide by 12.)}
\]

7. **Convert the probability to a percentage**:
\[
P(\text{two diamonds}) = \frac{1}{17} \approx 0.0588235294
\]
\[
P(\text{two diamonds}) \approx 0.0588 \text{ (in decimal form)}
\]
\[
\text{In percent form} = 0.0588 \times 100 = 5.88235294
\]

8. **Round to the nearest hundredth**:
- Rounding 5.88235294 yields 5.88%.

Thus, the probability of drawing two diamond cards is **5.88%**.