To calculate the probabilities of these scenarios from a standard 52-card deck, we need to understand basic combinatorial methods.
Part A: Probability of drawing 3 aces and 2 kings
First, let’s determine the number of ways to draw 3 aces and 2 kings. In a standard deck, there are 4 aces and 4 kings.
The number of ways to choose 3 aces from 4 is calculated using combinations:
C(4, 3) = 4
Next, the number of ways to choose 2 kings from 4 is:
C(4, 2) = 6
Now, we multiply those results to find the total number of favorable outcomes for this event:
4 (ways to choose aces) * 6 (ways to choose kings) = 24 favorable outcomes
Next, we need to find the total number of ways to draw 5 cards from the 52-card deck:
C(52, 5) = 2,598,960
Finally, the probability of drawing 3 aces and 2 kings is:
Probability = (Number of favorable outcomes) / (Total outcomes) = 24 / 2,598,960 ≈ 0.00000924
Part B: Probability of drawing a full house (3 cards of one kind and 2 cards of another kind)
To find the probability of getting a full house, we follow a similar approach:
First, choose the rank for the 3 cards. There are 13 possible ranks, and selecting one gives:
C(13, 1) = 13
Next, we choose 3 cards from the 4 available cards of that rank:
C(4, 3) = 4
Now, we need to choose a different rank for the 2 cards. There are 12 remaining ranks:
C(12, 1) = 12
Finally, we choose 2 cards from the 4 available cards of that second rank:
C(4, 2) = 6
The total number of favorable outcomes for a full house is calculated as follows:
13 (ways to choose the rank for 3 cards) * 4 (ways to choose 3 cards) * 12 (ways to choose a different rank for 2 cards) * 6 (ways to choose 2 cards) = 13 * 4 * 12 * 6 = 3,744 favorable outcomes
Using the same total number of outcomes from earlier for drawing 5 cards:
C(52, 5) = 2,598,960
Thus, the probability of drawing a full house is:
Probability = (Number of favorable outcomes) / (Total outcomes) = 3,744 / 2,598,960 ≈ 0.001440576
In summary, the probability of drawing 3 aces and 2 kings is about 0.00000924, and the probability of getting a full house is approximately 0.00144.