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)

Question

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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%.