Question

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

1. **Total Number of Diamonds**: There are 13 diamonds in the deck.
2. **Total Cards**: There are a total of 52 cards in the deck.

### Step 1: Probability of the first card being a diamond
The probability of drawing the first diamond is:
\[
P(\text{First diamond}) = \frac{\text{Number of diamonds}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4}
\]

### Step 2: Probability of the second card being a diamond
After drawing the first diamond, there are now 12 diamonds left and only 51 cards remaining in total. The probability of drawing a diamond for the second card is:
\[
P(\text{Second diamond | First diamond}) = \frac{\text{Remaining diamonds}}{\text{Remaining total cards}} = \frac{12}{51}
\]

### Step 3: Joint Probability
To find the probability of both events happening (drawing two diamonds), we multiply the probabilities:
\[
P(\text{Two diamonds}) = P(\text{First diamond}) \times P(\text{Second diamond | First diamond}) = \frac{13}{52} \times \frac{12}{51}
\]
\[
P(\text{Two diamonds}) = \frac{1}{4} \times \frac{12}{51} = \frac{12}{204} = \frac{1}{17}
\]

### Step 4: Converting to percentage
To express the answer in percent, we convert the fraction to a decimal and then multiply by 100:
\[
P(\text{Two diamonds}) \approx \frac{1}{17} \approx 0.058823529 \text{ (approx)}
\]
Now convert to percent:
\[
0.058823529 \times 100 \approx 5.8823529
\]
Rounding to the nearest hundredth gives:
\[
5.88\%
\]

**Final Answer**:
The probability of drawing two diamond cards is approximately **5.88%**.