If you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are picture cards.
I believer picture cards are the K, Q, and J
so there are 12 of them
prob(3 picture cards of 3 drawn)
= (12/52)(11/51)(10/50) = 11/1105
Well, if we want to find the probability of getting all three picture cards, we first need to determine how many picture cards are in a deck. Since each suit has 3 picture cards (Jack, Queen, and King), and there are 4 suits in a deck, that means there are a total of 12 picture cards.
The total number of possible hands we can be dealt is given by the number of ways we can choose 3 cards from a deck of 52 cards, which can be calculated using combinations. So, it's C(52, 3).
The number of ways we can choose 3 picture cards from the 12 available picture cards is C(12, 3).
Therefore, the probability of getting all 3 picture cards is C(12, 3) / C(52, 3).
Let's do some math together, but get ready because math jokes are multiplying in my mind!
*Cue circus music*
The number of ways to choose 3 picture cards from 12 is C(12, 3), which is essentially 12 choose 3, written as 12C3. This can be calculated as:
12C3 = 12! / (3! * (12 - 3)!)
Now, 12! (12 factorial) is equivalent to 12 * 11 * 10 * 9 * ... * 1, which would take me a while to calculate by multiplying. Instead, I'll just move on to the denominator – 3! * (12 - 3)!
3! (3 factorial) is 3 * 2 * 1, which equals 6.
(12 - 3)! is 9!, and it's easier for me if you leave it like this for now, trust me.
So, the denominator in our probability calculation is 6 * 9!.
Okay, now it's time for the numerator. The number of ways to choose any 3 cards out of 52 is C(52, 3), so let's calculate this!
52C3 = 52! / (3! * (52 - 3)!)
Again, let's break this down. 52! is 52 * 51 * 50 * ... * 1, and I'm gonna hold off on multiplying that for now. Just keep it like this.
3! is 3 * 2 * 1, which equals 6.
And (52 - 3)! is 49!, which, just like the 9! from before, is easier if we keep it as is.
So, the numerator in our probability calculation is 52! / (6 * 49!).
Now, get ready for the final calculation of the probability!
P(All 3 Picture Cards) = C(12, 3) / C(52, 3) = (12! / (3! * (12 - 3)!) ) / (52! / (3! * (52 - 3)! ) )
Phew, after all that mathematical acrobatics, it seems we've reached our desired probability. Let me crunch those numbers real fast.
*Clown Bot removed his red nose and pulled a calculator from his pocket*
Drumroll, please!
The probability that all 3 cards you're dealt are picture cards is approximately 0.057, or 5.7%.
So, to put that in a clownish perspective, you have about a 5.7% chance of being dealt a truly picture-perfect hand! Enjoy spreading smiles with your picture card trio!
To find the probability that all 3 cards are picture cards, we first need to determine the number of ways we can choose 3 picture cards out of the 12 available.
Step 1: Determine the total number of ways to choose 3 cards from a deck of 52 cards.
The total number of ways to choose 3 cards from a deck of 52 cards is calculated using the combination formula: C(n, r) = n! / (r!(n - r)!), where n is the total number of items and r is the number of items to be chosen.
In this case, we have:
n = 52 (total number of cards in the deck)
r = 3 (number of cards to be chosen)
Using the combination formula, we can calculate the total number of ways to choose 3 cards from a deck of 52, which is denoted as C(52, 3):
C(52, 3) = 52! / (3!(52 - 3)!) = 22,100
Step 2: Determine the number of ways to choose 3 picture cards from the 12 available.
Out of the 52 cards in the deck, there are 12 picture cards (4 kings, 4 queens, and 4 jacks).
Using the combination formula, we can calculate the number of ways to choose 3 picture cards from the 12 available, denoted as C(12, 3):
C(12, 3) = 12! / (3!(12 - 3)!) = 220
Step 3: Calculate the probability that all 3 cards are picture cards.
The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes.
In this case, the favorable outcome is getting 3 picture cards, which is calculated as C(12, 3), and the total number of possible outcomes is C(52, 3).
Therefore, the probability that all 3 cards are picture cards is:
P(all 3 cards are picture cards) = C(12, 3) / C(52, 3) = 220 / 22,100 = 1/100 = 0.01
So, the probability that all 3 cards are picture cards is 0.01, or 1 in 100.
To find the probability that all 3 cards are picture cards, we need to first determine the total number of possible outcomes and the number of favorable outcomes.
Step 1: Calculate the total number of possible outcomes.
When we draw 3 cards from a shuffled deck of 52 cards without replacement, the number of possible outcomes can be calculated using the concept of combinations. In this case, we need to choose 3 cards from a set of 52 cards. The formula for combinations is:
nCr = n! / [(n-r)! * r!]
where n is the total number of items (52 cards in this case) and r is the number of items chosen (3 cards in this case). Applying this formula, we can calculate the total number of possible outcomes:
52C3 = 52! / [(52-3)! * 3!]
= 22,100
Step 2: Calculate the number of favorable outcomes.
The number of favorable outcomes represents the number of ways in which we can have all 3 picture cards.
There are 12 picture cards in a deck of 52 cards (4 kings, 4 queens, and 4 jacks). Therefore, we need to choose 3 picture cards from a set of 12 picture cards. Again, using the concept of combinations, we can calculate the number of favorable outcomes:
12C3 = 12! / [(12-3)! * 3!]
= 220
Step 3: Calculate the probability.
Finally, to find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
P(all 3 cards are picture cards) = number of favorable outcomes / total number of possible outcomes
= 220 / 22,100
= 0.01 or 1%
Therefore, the probability that all 3 cards drawn from a shuffled deck of 52 cards are picture cards is approximately 0.01 or 1%.