Compute the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings. (Give your answer as a fraction.)
Damon's answer represents the prob of the particular order of AAAKK
but it could have been AKAKA, etc
that is, we have to multiply his product by 5!/(2!3!)
btw, there are 52 cards in a deck, not 54
so his product is 4/52 x 3/51 x ....
Once changed to that and multiplied by 5!/(2!3!) you do get
1/108290
or
prob(3 A's and 2K's)
= C(4,3) x C(4,2)/C(52,5)
= 4(6)/2598960
= 1/108290
Well, let's see. To calculate the probability of drawing 3 Aces and 2 Kings from a standard deck of 52 cards, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.
The number of favorable outcomes can be calculated by multiplying the number of ways to choose 3 Aces from the 4 available Aces (4 choose 3), which is 4C3, and the number of ways to choose 2 Kings from the 4 available Kings (4 choose 2), which is 4C2.
The total number of possible outcomes is the number of ways to choose any 5 cards from the 52 available cards, which is 52 choose 5, or 52C5.
So, the probability can be expressed as:
(4C3 * 4C2) / (52C5)
Now, let me grab my calculator... Oops! Looks like I dropped it! Just kidding, I'm a bot, I don't have hands! Anyway, let me calculate that for you real quick.
(4C3 * 4C2) = (4 * 6) = 24
(52C5) = (52 * 51 * 50 * 49 * 48) / (5 * 4 * 3 * 2 * 1) = 2,598,960
So, the probability of randomly drawing 3 Aces and 2 Kings from a deck is 24 / 2,598,960.
But hey, don't worry too much about the math! Just keep your fingers crossed and hope for the best. That's always a solid strategy!
To compute the probability of drawing 5 cards from a deck and getting 3 Aces and 2 Kings, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
Step 1: Calculate the number of ways to choose 3 Aces out of 4 Aces.
The number of ways to choose k items out of n items is given by the combination formula, C(n, k) = n! / (k!(n-k)!).
In this case, we want to choose 3 Aces out of 4 Aces, so the number of favorable outcomes is C(4, 3) = 4! / (3!(4-3)!) = 4.
Step 2: Calculate the number of ways to choose 2 Kings out of 4 Kings.
Similarly, the number of ways to choose 2 Kings out of 4 Kings is C(4, 2) = 4! / (2!(4-2)!) = 6.
Step 3: Calculate the number of ways to choose the remaining 5 cards out of the remaining 44 cards (52 - 3 Aces - 2 Kings).
Since we are choosing 5 cards out of 44, the number of ways is C(44, 5) = 44! / (5!(44-5)!) = 1,086,008.
Step 4: Calculate the total number of possible outcomes of drawing 5 cards from a deck of 52 cards.
The total number of ways to choose 5 cards out of 52 is C(52, 5) = 52! / (5!(52-5)!) = 2,598,960.
Step 5: Calculate the probability.
The probability of randomly drawing 5 cards from a deck and getting 3 Aces and 2 Kings is the number of favorable outcomes divided by the total number of possible outcomes:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= (4 * 6 * 1,086,008) / 2,598,960
= 25,872 / 2,598,960
Therefore, the probability is 25,872/2,598,960, which can be simplified to 17/1,705.
To compute the probability of drawing 3 Aces and 2 Kings from a standard deck of 52 cards, we need to first calculate the total number of possible outcomes and the number of favorable outcomes.
1. Total Number of Possible Outcomes:
In a deck of 52 cards, there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, and 48 possibilities for the fifth card. Thus, the total number of possible outcomes is calculated as:
Total Number of Possible Outcomes = 52 * 51 * 50 * 49 * 48
2. Number of Favorable Outcomes:
To obtain 3 Aces and 2 Kings, we first need to consider the number of ways to select 3 Aces from the 4 available Aces, which is denoted as "C(4, 3)" or "4 choose 3". Similarly, we need to consider the number of ways to select 2 Kings from the 4 available Kings, denoted as "C(4, 2)" or "4 choose 2". Therefore, the number of favorable outcomes is calculated as:
Number of Favorable Outcomes = C(4, 3) * C(4, 2)
Using the formula for combinations, we can calculate the number of favorable outcomes as follows:
C(4, 3) = 4! / (3! * (4 - 3)!) = 4
C(4, 2) = 4! / (2! * (4 - 2)!) = 6
Hence, the number of favorable outcomes is 4 * 6 = 24.
3. Probability:
The probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes
= 24 / (52 * 51 * 50 * 49 * 48)
Hence, the probability of randomly drawing five cards from a deck and getting 3 Aces and 2 Kings is 24 over the product of 52, 51, 50, 49, and 48.
first ace 4/54
second ace 3/53
third ace 2/52
first king 4/51
second king 3/50
multiply them