Three cards are drawn from a deck of 52 cards . What is the probability that all of them are diamonds

Answer

To solve the problem of finding the probability that all three cards drawn from a deck of 52 cards are diamonds, we can follow these steps:

### Step 1: Identify the total number of cards and the number of diamonds
In a standard deck of cards, there are 52 cards in total, and among them, there are 13 diamonds.

### Step 2: Determine the total outcomes for drawing 3 cards
The total number of ways to draw 3 cards from 52 cards can be calculated using the combination formula:
\[ \text{Total outcomes} = \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52!}{3! \cdot 49!} \]

### Step 3: Determine the favorable outcomes for drawing 3 diamonds
The number of ways to draw 3 diamonds from the 13 available diamonds is also calculated using the combination formula:
\[ \text{Favorable outcomes} = \binom{13}{3} = \frac{13!}{3!(13-3)!} = \frac{13!}{3! \cdot 10!} \]

### Step 4: Calculate the probability
The probability \( P \) that all three drawn cards are diamonds is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{all diamonds}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{\binom{13}{3}}{\binom{52}{3}} \]

### Step 5: Substitute the values into the formulas
Calculating the values:
1. Calculate \( \binom{52}{3} \):
\[ \binom{52}{3} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = \frac{132600}{6} = 22100 \]

2. Calculate \( \binom{13}{3} \):
\[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = \frac{1716}{6} = 286 \]

### Step 6: Final calculation of the probability
Now substituting these values into the probability formula:
\[ P(\text{all diamonds}) = \frac{286}{22100} \]

### Step 7: Simplify the fraction if possible
The fraction \( \frac{286}{22100} \) can be simplified:
\[ P(\text{all diamonds}) \approx 0.01295 \]

### Conclusion
Thus, the probability that all three cards drawn are diamonds is approximately \( 0.01295 \) or \( \frac{143}{11050} \) when simplified.

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