if you are dealt 5 cards from a shuffled deck of 52 cards find the probability that all 5 cards are picture cards please show how to get the answer such 6!C! so i can understand
12^C 5/ 52^C 5 = 33/108290
Well, let's figure out the probability of getting all 5 picture cards from a shuffled deck of 52 cards.
First, we need to find the total number of ways to choose 5 cards from a deck of 52. This can be calculated using the combination formula, which is nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items selected.
In this case, n is 52 (the total number of cards in a deck) and r is 5 (the number of cards we want to select).
So, the number of ways to select 5 cards from a deck of 52 is 52C5 = 52! / (5! * (52-5)!).
Next, we need to find the number of ways to choose 5 picture cards from the deck. In a standard deck, there are 12 picture cards (4 Kings, 4 Queens, and 4 Jacks).
So, we need to calculate the number of ways to choose 5 cards out of these 12 picture cards. This can be done using the combination formula as well:
12C5 = 12! / (5! * (12-5)!)
Finally, to find the probability, we divide the number of ways to choose 5 picture cards by the total number of ways to choose any 5 cards:
Probability = (12C5) / (52C5) = ((12! / (5! * (12-5)!)) / (52! / (5! * (52-5)!)))
Simplifying further:
Probability = (12! * (52-5)!) / (5! * (12-5)! * 52!)
Probability ≈ 0.00144
So, the probability of getting all 5 picture cards from a shuffled deck of 52 cards is approximately 0.00144, or roughly 0.144%.
To find the probability of getting all 5 picture cards when dealt 5 cards from a shuffled deck of 52 cards, we first need to determine the number of favorable outcomes (the number of ways to get 5 picture cards) and the total number of possible outcomes (the total number of ways to choose any 5 cards from the deck).
1. Determine the number of ways to get 5 picture cards:
There are 12 picture cards (4 kings, 4 queens, and 4 jacks) in a deck of 52 cards. Since we need all 5 cards to be picture cards, we need to select 5 cards out of these 12 picture cards.
To calculate this, we use combination formula:
nCr = n! / (r! * (n-r)!)
So, the number of ways to choose 5 picture cards from 12 is:
12C5 = 12! / (5! * (12-5)!)
2. Determine the total number of possible outcomes:
The total number of ways to choose 5 cards from a deck of 52 cards can be calculated using the same combination formula.
So, the total number of ways to choose any 5 cards from 52 is:
52C5 = 52! / (5! * (52-5)!)
3. Calculate the probability:
The probability is equal to the number of favorable outcomes divided by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 12C5 / 52C5
Now, substituting the values:
Probability = (12! / (5! * (12-5)!) / (52! / (5! * (52-5)!))
Probability = (12! * (52-5)! * (5!) * (5!)) / ((5!) * (7!) * (52! - 5!) * (5!))
Probability = (12! * (52-5)! * (5!)) / ((7!) * (52! - 5!))
Simplifying the factorials:
Probability = (12! * 47!) / (7! * 52!)
Probability = 12 * 11 * 10 * 9 * 8 / 52 * 51 * 50 * 49 * 48
Calculating the result:
Probability = 0.00144 (rounded to 5 decimal places)
Therefore, the probability of getting all 5 picture cards is approximately 0.00144.
Question:
Do you consider an ace a "picture" card?
I will assume you only want the J, Q, and K
There are 12 of those
So the number of sets of 5 form those 12 = C(12,5)
Without restriction, the number of sets of 5 is C(52,5)
so the prob that your 5 cards are "picture" cards
= C(12,5)/C(52,5) = 792/259860 = .000305
If your interpretation is otherwise make the necessary changes.