What is the probability of being dealt 3 aces and 2 kings, and what is the probability of being dealt a full house (3 cards of one kind and 2 cards of another kind) from a standard 52-card deck?

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To calculate the probabilities, we first need to understand the total number of possible 5-card combinations that can be drawn from a standard 52-card deck. This can be calculated using the combination formula:

Total combinations: C(52, 5) = 2,598,960

1. Probability of Drawing 3 Aces and 2 Kings

To draw 3 aces and 2 kings, we need to calculate the number of favorable combinations:

  1. Choosing 3 Aces: There are 4 aces in the deck. The number of ways to choose 3 aces from 4 is:

    2. C(4, 3) = 4

  2. Choosing 2 Kings: Similarly, there are also 4 kings in the deck. The number of ways to choose 2 kings from 4 is:

    3. C(4, 2) = 6

Now, we can find the total ways to choose 3 aces and 2 kings:

Total combinations for 3 Aces and 2 Kings: 4 * 6 = 24

Calculating Probability:

The probability of drawing 3 aces and 2 kings is:

P(3 Aces, 2 Kings) = Favorable Combinations / Total Combinations

P(3 Aces, 2 Kings) = 24 / 2,598,960 = 0.00000924

2. Probability of Drawing a Full House (3 Cards of One Kind and 2 Cards of Another Kind)

For a full house, we need to select 3 cards of one rank and 2 cards of another rank.

  1. Choosing the Rank for 3 Cards: There are 13 ranks in total. Selecting one rank to have 3 cards can be done in:

    2. C(13, 1) = 13

  2. Choosing the Cards: For the chosen rank, there are 4 cards available. The number of ways to choose 3 out of these 4 cards is:

    3. C(4, 3) = 4

  3. Choosing the Rank for 2 Cards: For the second rank (which has to be different from the first), we have 12 ranks to choose from:

    4. C(12, 1) = 12

  4. Choosing the Cards: From this chosen rank, we can select 2 cards from 4. The number of ways is:

    5. C(4, 2) = 6

The total combinations for a full house is:

Total combinations for Full House: 13 * 4 * 12 * 6 = 3,744

Calculating Probability:

The probability of drawing a full house is:

P(Full House) = Favorable Combinations / Total Combinations

P(Full House) = 3,744 / 2,598,960 = 0.001440576

Final Probabilities:

  • Probability of drawing 3 Aces and 2 Kings: 0.00000924
  • Probability of drawing a Full House: 0.001440576

In summary, while the chance of drawing 3 aces and 2 kings is astronomically low, the probability of drawing a full house is considerably higher, yet still a rare event when dealing 5 cards from a standard deck.

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