Compute the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings. (Give your answer as a fraction.)
Probability
To compute the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings, we first need to determine the number of favorable outcomes and the total number of possible outcomes.
First, let's find the number of favorable outcomes:
To have 3 Aces and 2 Kings, we need to choose 3 out of the 4 Aces and 2 out of the 4 Kings. We can do this using combinations.
The number of ways to choose 3 Aces from 4 is given by:
C(n, k) = n! / (k!(n-k)!)
C(4, 3) = 4! / (3!(4-3)!)
= 4! / (3!1!)
= 4
Similarly, the number of ways to choose 2 Kings from 4 is given by:
C(n, k) = n! / (k!(n-k)!)
C(4, 2) = 4! / (2!(4-2)!)
= 4! / (2!2!)
= 6
Therefore, the number of favorable outcomes is 4 * 6 = 24.
Now, let's find the total number of possible outcomes:
When drawing 5 cards from a standard deck of 52 cards, the total number of possible outcomes is given by:
C(n, k) = n! / (k!(n-k)!)
C(52, 5) = 52! / (5!(52-5)!)
= 52! / (5!47!)
≈ 2,598,960
Therefore, the total number of possible outcomes is 2,598,960.
Now we can calculate the probability of getting 3 Aces and 2 Kings:
Probability = Favorable outcomes / Total outcomes
= 24 / 2,598,960
= 1 / 108,290
Hence, 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 possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
When drawing five cards from a standard deck of 52 cards, the number of possible outcomes is given by the combinations formula:
C(52, 5) = 52! / (5! * (52-5)!) = 52! / (5! * 47!) = 2,598,960.
Number of favorable outcomes:
To get 3 Aces and 2 Kings, we need to choose 3 Aces from the 4 available and 2 Kings from the remaining 4. Therefore, the number of favorable outcomes is given by the product of these two combinations:
C(4, 3) * C(4, 2) = (4! / (3! * (4-3)!) * 4! / (2! * (4-2)!) = 4 * 6 = 24.
Probability:
The probability of obtaining 3 Aces and 2 Kings can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P = Number of favorable outcomes / Total number of possible outcomes
P = 24 / 2,598,960
Therefore, the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings is:
P = 1 / 108,290