Compute the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings. (Give your answer as a fraction.)
To compute the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings, we need to calculate the ratio of favorable outcomes to the total possible outcomes.
Step 1: Calculate the number of ways to choose 3 Aces from a deck of 4 Aces:
C(4, 3) = 4! / (3! * (4-3)!) = 4
Step 2: Calculate the number of ways to choose 2 Kings from a deck of 4 Kings:
C(4, 2) = 4! / (2! * (4-2)!) = 6
Step 3: Calculate the number of ways to choose the remaining 5 - 3 - 2 = 0 cards from the remaining 52 - 4 - 4 = 44 cards in the deck:
C(44, 0) = 44! / (0! * (44-0)!) = 1
Step 4: Calculate the total possible outcomes of choosing 5 cards from a deck of 52 cards:
C(52, 5) = 52! / (5! * (52-5)!) = 2,598,960
Step 5: Calculate the product of the number of ways in steps 1, 2, and 3:
4 * 6 * 1 = 24
Step 6: Calculate the probability by dividing the product of step 5 by the total number of possible outcomes in step 4:
P = 24 / 2,598,960 = 1 / 108,290
Therefore, the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings is 1/108,290.
To compute the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings, we need to determine the total number of favorable outcomes (getting 3 Aces and 2 Kings) and divide it by the total number of possible outcomes (all combinations of 5 cards from a deck).
First, let's determine the total number of favorable outcomes, which is the number of ways to choose 3 Aces from the 4 Aces in the deck multiplied by the number of ways to choose 2 Kings from the 4 Kings in the deck.
The number of ways to choose 3 Aces from 4 is given by the binomial coefficient: C(4, 3) = 4! / (3! * (4-3)!) = 4.
Similarly, the number of ways to choose 2 Kings from 4 is: C(4, 2) = 4! / (2! * (4-2)!) = 6.
Now, we multiply these values together to get the total number of favorable outcomes: 4 * 6 = 24.
Next, we need to determine the total number of possible outcomes, which is the number of ways to choose any 5 cards from the 52-card deck. This is given by the binomial coefficient: C(52, 5) = 52! / (5! * (52-5)!) = 2,598,960.
Finally, we divide the number of favorable outcomes by the number of possible outcomes to calculate the probability:
P(3 Aces and 2 Kings) = 24 / 2,598,960.
Hence, the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings is 24/2,598,960, which can be simplified but cannot be expressed as a simple fraction.