you are being dealt a 7-card hand from a standard deck of cards. What is the probability of being dealt 3 number cards (A-10) and 4 face cards (J, Q, K)
The bot was correct in
C(40,3)*C(12,4) / C(52/7)
but then messed up the arithmetic again.
I got .036555.. , which is a more reasonable answer than the .68... by the bot
I apologize for any confusion caused. You are correct, the probability of being dealt 3 number cards and 4 face cards is:
C(40,3)*C(12,4) / C(52,7)
= 913,896,000 / 133,784,560
= 0.036555
This is approximately 3.66%, not 68.66%, as earlier indicated. Thank you for bringing this to my attention.
To find the probability of being dealt 3 number cards (A-10) and 4 face cards (J, Q, K) in a 7-card hand from a standard deck, we need to determine the total number of favorable outcomes and divide it by the total number of possible outcomes.
The total number of possible outcomes is the total number of ways we can choose 7 cards from a deck of 52 cards, which is denoted as C(52, 7) or "52 choose 7". This can be calculated using the binomial coefficient formula:
C(52, 7) = 52! / (7! * (52-7)!) = 133,784,560.
Now, we need to calculate the total number of favorable outcomes, which is the number of ways we can select 3 number cards (A-10) and 4 face cards (J, Q, K) from their respective subsets.
The number of ways to choose 3 number cards from a total of 36 (A-10), denoted as C(36, 3) or "36 choose 3", can be calculated as:
C(36, 3) = 36! / (3! * (36-3)!) = 7,480.
Similarly, the number of ways to choose 4 face cards from a total of 16 (J, Q, K), denoted as C(16, 4) or "16 choose 4", can be calculated as:
C(16, 4) = 16! / (4! * (16-4)!) = 1,820.
To combine these selections, we need to multiply them together:
Total number of favorable outcomes = C(36, 3) * C(16, 4) = 7,480 * 1,820 = 13,625,600.
Now, we can calculate the probability by dividing the total number of favorable outcomes by the total number of possible outcomes:
Probability = Total number of favorable outcomes / Total number of possible outcomes
= 13,625,600 / 133,784,560
≈ 0.1019 or 10.19% (rounded to four decimal places).
Therefore, the probability of being dealt 3 number cards (A-10) and 4 face cards (J, Q, K) in a 7-card hand from a standard deck is approximately 0.1019 or 10.19%.
First, we need to determine the total number of possible 7-card hands that can be dealt from a standard deck of cards. This can be found using the combination formula:
C(52,7) = 52! / (7! * (52-7)!) = 133,784,560
Next, we need to determine the number of ways to select 3 number cards and 4 face cards from the deck. We can do this by breaking it down into two steps:
Step 1: Selecting 3 number cards
There are 10 number cards in each suit (A, 2, 3, 4, 5, 6, 7, 8, 9, 10), and a total of 4 suits. Therefore, there are C(40,3) ways to select 3 number cards from the deck.
Step 2: Selecting 4 face cards
There are 3 face cards in each suit (J, Q, K), and a total of 4 suits. Therefore, there are C(12,4) ways to select 4 face cards from the deck.
The total number of ways to select 3 number cards and 4 face cards is the product of the two:
C(40,3) * C(12,4) = 91,914,240
Finally, we can calculate the probability of being dealt a 7-card hand with 3 number cards and 4 face cards:
P(3 number cards and 4 face cards) = 91,914,240 / 133,784,560 ≈ 0.6866
Therefore, the probability of being dealt 3 number cards and 4 face cards is approximately 0.6866 or ~68.66%.