To calculate the probability that a hand of five cards drawn at random from a deck of 52 playing cards contains exactly:

• 2 Kings,
• 2 Queens,
• and no Jacks,

we can break this problem down step by step.

Step 1: Number of possible hands

The total number of ways to choose 5 cards from a deck of 52 is given by the combination formula:

$\text{Total number of possible hands} = \binom{52}{5}$

$\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960$

Step 2: Number of favorable hands

We now need to find the number of hands that meet the specific criteria of having 2 Kings, 2 Queens, and no Jacks.

2 Kings:

There are 4 Kings in the deck, and we need to choose 2 of them. The number of ways to do this is:

$\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$

2 Queens:

There are 4 Queens in the deck, and we need to choose 2 of them. The number of ways to do this is:

$\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$

1 other card (no Jack):

We need to choose 1 more card that is neither a King, a Queen, nor a Jack. There are 4 Jacks in the deck, so the remaining cards (besides Kings and Queens) are 52 - 4 Kings - 4 Queens - 4 Jacks = 40 cards.

The number of ways to choose 1 card from these 40 is:

$\binom{40}{1} = 40$

Step 3: Calculate the number of favorable hands

The total number of favorable hands is the product of the individual combinations:

$\text{Number of favorable hands} = \binom{4}{2} \times \binom{4}{2} \times \binom{40}{1} = 6 \times 6 \times 40 = 1440$

Step 4: Calculate the probability

Finally, the probability is the ratio of favorable hands to total possible hands:

$\text{Probability} = \frac{1440}{2,598,960} \approx 0.000554$

Thus, the probability of drawing exactly 2 Kings, 2 Queens, and no Jacks in a hand of 5 cards is approximately $0.000554$ or $0.0554\%$.

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Do you want any more details or have any questions?

Here are 5 related questions for you to consider:

1. What is the probability of drawing 3 Kings, 1 Queen, and no Jacks?
2. How does the probability change if we draw 4 cards instead of 5?
3. What is the probability of drawing exactly one King and no Queens or Jacks?
4. How would the probability change if we were allowed to replace the cards in the deck?
5. What is the probability of drawing a hand with no face cards (Kings, Queens, or Jacks)?

Tip: When calculating probabilities in card games, always remember to account for the different categories of cards and whether replacement affects the outcomes.