Three cards are drawn from a deck of 52 cards . What is the probability that
they are all red cards ?

Answer

To find the probability that all three cards drawn from a deck of 52 cards are red, we can follow these steps:

### Step 1: Identify the total number of cards and the number of red cards.
- A standard deck has 52 cards.
- Out of these, 26 cards are red (13 hearts and 13 diamonds).

### Step 2: Determine the total number of ways to draw 3 cards from the deck.
- The total number of ways to choose 3 cards from 52 is given by the combination formula \( \binom{n}{r} \), which is calculated as:
\[ \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = 22100 \]

### Step 3: Determine the number of ways to draw 3 red cards.
- The number of ways to choose 3 red cards from the 26 red cards is:
\[ \binom{26}{3} = \frac{26!}{3!(26-3)!} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600 \]

### Step 4: Calculate the probability.
- The probability \( P \) that all three drawn cards are red is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of ways to choose 3 red cards}}{\text{Total number of ways to choose 3 cards}} = \frac{\binom{26}{3}}{\binom{52}{3}} = \frac{2600}{22100} \]

### Step 5: Simplify the probability.
- To simplify \( \frac{2600}{22100} \):
\[ P = \frac{2600 \div 1300}{22100 \div 1300} = \frac{2}{17} \]

### Final Answer:
The probability that all three cards drawn are red is \( \frac{2}{17} \).

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