Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is

Understanding Card Probability

This problem asks us to find the probability of drawing a specific set of three cards – a king, a queen, and a jack – from a standard pack of 52 cards. To solve this, we need to understand the concepts of combinations and probability.

Probability is calculated as the ratio of favorable outcomes to the total possible outcomes:

$$\text{Probability}=\frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

Total Possible Outcomes for Drawing Cards

First, let's determine the total number of ways to draw 3 cards from a pack of 52 cards. Since the order in which the cards are drawn does not matter, this is a combination problem. The formula for combinations is:

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$$

Where $n$ is the total number of items to choose from, and $k$ is the number of items to choose.

- Total cards in the pack, $n=52$
- Number of cards to draw, $k=3$

So, the total number of ways to draw 3 cards from 52 cards is:

$$\text{Total Outcomes}=\left(\genfrac{}{}{0ex}{}{52}{3}\right)=\frac{52!}{3!(52-3)!}$$

$$\text{Total Outcomes}=\frac{52\times 51\times 50}{3\times 2\times 1}$$

Let's perform the calculation:

- $52\times 51\times 50=132600$
- $3\times 2\times 1=6$
- $$\text{Total Outcomes}=\frac{132600}{6}=22100$$

Therefore, there are 22,100 total possible outcomes when drawing 3 cards from a 52-card deck.

Favorable Outcomes for Drawing Specific Cards

Next, we need to find the number of favorable outcomes, which is drawing one king, one queen, and one jack. A standard pack of 52 cards has:

- 4 kings
- 4 queens
- 4 jacks

We need to choose 1 king from 4, 1 queen from 4, and 1 jack from 4. These are independent selections.

- Number of ways to choose 1 king from 4 = $\left(\genfrac{}{}{0ex}{}{4}{1}\right)=4$
- Number of ways to choose 1 queen from 4 = $\left(\genfrac{}{}{0ex}{}{4}{1}\right)=4$
- Number of ways to choose 1 jack from 4 = $\left(\genfrac{}{}{0ex}{}{4}{1}\right)=4$

To find the total number of favorable outcomes, we multiply these possibilities:

$$\text{Favorable Outcomes}=\left(\genfrac{}{}{0ex}{}{4}{1}\right)\times \left(\genfrac{}{}{0ex}{}{4}{1}\right)\times \left(\genfrac{}{}{0ex}{}{4}{1}\right)$$

$$\text{Favorable Outcomes}=4\times 4\times 4=64$$

So, there are 64 favorable outcomes for drawing one king, one queen, and one jack.

Final Probability Calculation

Now, we can calculate the probability by dividing the favorable outcomes by the total possible outcomes:

$$\text{Probability}=\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$$

$$\text{Probability}=\frac{64}{22100}$$

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor. Both numbers are divisible by 4:

- $64÷4=16$
- $22100÷4=5525$

So, the simplified probability is:

$$\text{Probability}=\frac{16}{5525}$$

Conclusion

The probability of drawing a king, a queen, and a jack when three cards are drawn from a pack of 52 cards is $\frac{16}{5525}$.