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)

_______%

1 answer

To calculate the probability of drawing two diamond cards from a standard deck of 52 cards, we can break it down step by step:

  1. Total cards in the deck: There are 52 cards in total.
  2. Total diamond cards: There are 13 diamond cards in the deck.

Probability of drawing the first diamond card: When you draw the first card, there are 13 diamond cards out of 52 total cards. Thus, the probability of drawing a diamond initially is:

\[ P(\text{first diamond}) = \frac{13}{52} \]

Probability of drawing the second diamond card: After drawing one diamond card, there are now 12 diamond cards left and 51 cards remaining in total. Therefore, the probability of drawing a second diamond card after the first is:

\[ P(\text{second diamond | first diamond}) = \frac{12}{51} \]

Calculating the combined probability: To find the overall probability of both events happening (drawing two diamonds in a row), we multiply the two probabilities:

\[ P(\text{two diamonds}) = P(\text{first diamond}) \times P(\text{second diamond | first diamond}) = \frac{13}{52} \times \frac{12}{51} \]

Calculating this gives:

\[ P(\text{two diamonds}) = \frac{13}{52} \times \frac{12}{51} = \frac{13 \times 12}{52 \times 51} = \frac{156}{2652} \]

Now, simplifying \(\frac{156}{2652}\):

\[ \frac{156}{2652} = \frac{1}{17} \quad (\text{since } 156 \div 156 = 1 \text{ and } 2652 \div 156 = 17) \]

Next, we convert this probability into a percentage:

\[ \text{Percentage} = \left(\frac{1}{17}\right) \times 100 \approx 5.88235294117647 \]

Rounding this to the nearest hundredth:

\[ \text{Percentage} \approx 5.88% \]

Therefore, the final answer is:

\[ \boxed{5.88%} \]