If three cards are drawn from a pack of 52 cards, what is the chance that all will be queen?

Answer

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

### Step 1: Determine the Total Number of Ways to Draw 3 Cards
The total number of ways to draw 3 cards from a pack of 52 cards can be calculated using the combination formula:

\[ \text{Total ways} = \binom{52}{3} \]

### Step 2: Determine the Number of Favorable Outcomes
Next, we need to determine how many ways we can draw 3 queens from the 4 queens available in the deck. This can also be calculated using the combination formula:

\[ \text{Favorable ways} = \binom{4}{3} \]

### Step 3: Calculate the Probability
The probability of drawing 3 queens is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P(\text{all queens}) = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{\binom{4}{3}}{\binom{52}{3}} \]

### Step 4: Calculate the Combinations
Now we calculate the combinations:

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 \(\binom{4}{3}\):
\[ \binom{4}{3} = \frac{4}{1} = 4 \]

### Step 5: Substitute the Values into the Probability Formula
Now we substitute the values we calculated into the probability formula:

\[ P(\text{all queens}) = \frac{4}{22100} \]

### Step 6: Simplify the Probability
The final probability can be simplified if needed, but in this case, we can leave it as:

\[ P(\text{all queens}) = \frac{4}{22100} \approx 0.000181 \]

Thus, the probability that all three cards drawn will be queens is \(\frac{4}{22100}\).

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