Four cards are randomly drawn from a standard deck of 52 cards. Find each probability.
a. P(1 ace and 3 kings)
b. P(2 odd and 2 face cards)
a. P(1 ace and 3 kings)
Well, I don't have a deck of cards with me right now, but let me whip out my trusty deck of imaginary cards to help me out. Ah, here we go.
There are 4 aces in a standard deck and 4 kings. So the probability of drawing 1 ace and 3 kings is equal to the number of ways you can choose 1 ace from 4 aces multiplied by the number of ways you can choose 3 kings from 4 kings, divided by the total number of possible outcomes.
Let's do the math:
Number of ways to choose 1 ace from 4 = C(4, 1) = 4
Number of ways to choose 3 kings from 4 = C(4, 3) = 4
Total number of possible outcomes = C(52, 4) = 270,725
So, the probability P(1 ace and 3 kings) = (4 * 4) / 270,725
Now, if only I had a deck of cards and a calculator handy, I could give you the exact answer. But life is full of mysteries, isn't it?
b. P(2 odd and 2 face cards)
Ah, a mixture of odd and face cards? Sounds like a party I'd like to be invited to!
In a standard deck, there are 26 red cards and 26 black cards, with half of each being odd cards. And there are 12 face cards (Jack, Queen, King) of each suit, totaling to 48 face cards.
To find the probability of drawing 2 odd cards and 2 face cards, we need to calculate the number of ways to choose 2 odd cards from 26 and 2 face cards from 48. Then we'll divide that by the total number of possible outcomes.
Let's do the math:
Number of ways to choose 2 odd cards from 26 = C(26, 2) = 325
Number of ways to choose 2 face cards from 48 = C(48, 2) = 1,128
Total number of possible outcomes = C(52, 4) = 270,725
So, the probability P(2 odd and 2 face cards) = (325 * 1,128) / 270,725
Well, there you have it! I hope my imaginary deck of cards and my math skills were entertaining enough for you.
a. To find the probability of drawing 1 ace and 3 kings, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
Number of favorable outcomes:
There is only 1 ace and 4 kings in a standard deck of 52 cards. Therefore, the number of ways to choose 1 ace and 3 kings is 1 * (4 choose 3) = 4.
Total number of possible outcomes:
Since we are drawing 4 cards from a deck of 52 cards, the total number of possible outcomes is (52 choose 4) = C(52, 4) = 270,725.
Probability = Number of favorable outcomes / Total number of possible outcomes
P(1 ace and 3 kings) = 4 / 270,725
b. To find the probability of drawing 2 odd cards and 2 face cards, we again need to calculate the number of favorable outcomes and the total number of possible outcomes.
Number of favorable outcomes:
There are 20 odd cards and 12 face cards in a standard deck of 52 cards. Therefore, the number of ways to choose 2 odd cards and 2 face cards is (20 choose 2) * (12 choose 2) = C(20, 2) * C(12, 2) = 190 * 66 = 12,540.
Total number of possible outcomes:
We are drawing 4 cards from a deck of 52 cards, so the total number of possible outcomes is (52 choose 4) = C(52, 4) = 270,725.
Probability = Number of favorable outcomes / Total number of possible outcomes
P(2 odd and 2 face cards) = 12,540 / 270,725
To find the probabilities, we need to calculate the total number of favorable outcomes (the desired combinations) and divide it by the total number of possible outcomes (all the combinations that can occur).
a. P(1 ace and 3 kings):
First, let's find the total number of ways to choose 4 cards from a deck of 52 cards, which can be calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of cards (52) and r is the number of cards we want to draw (4).
C(52, 4) = 52! / (4!(52-4)!) = 52! / (4! * 48!) = (52 * 51 * 50 * 49) / (4 * 3 * 2 * 1) = 270,725
Now, let's find the number of ways to select 1 ace and 3 kings from the deck:
Number of ways to select 1 ace = C(4, 1) = 4
Number of ways to select 3 kings = C(4, 3) = 4
Since these events are independent, we multiply the number of ways to select each combination:
Number of favorable outcomes = (Number of ways to select 1 ace) * (Number of ways to select 3 kings) = 4 * 4 = 16
Now, we can calculate the probability:
P(1 ace and 3 kings) = (Number of favorable outcomes) / (Total number of outcomes) = 16 / 270,725 ≈ 0.000059
Therefore, the probability of drawing 1 ace and 3 kings is approximately 0.000059.
b. P(2 odd and 2 face cards):
Similarly, let's find the total number of ways to choose 4 cards from a deck of 52 cards:
C(52, 4) = 52! / (4!(52-4)!) = 52! / (4! * 48!) = 270,725
Now, let's find the number of ways to select 2 odd cards and 2 face cards from the deck:
Number of ways to select 2 odd cards = C(26, 2) = 26! / (2!(26-2)!) = (26 * 25) / (2 * 1) = 325
Number of ways to select 2 face cards = C(12, 2) = 12! / (2!(12-2)!) = (12 * 11) / (2 * 1) = 66
Again, since these events are independent, we multiply the number of ways to select each combination:
Number of favorable outcomes = (Number of ways to select 2 odd cards) * (Number of ways to select 2 face cards) = 325 * 66 = 21,450
Now, we can calculate the probability:
P(2 odd and 2 face cards) = (Number of favorable outcomes) / (Total number of outcomes) = 21,450 / 270,725 ≈ 0.079
Therefore, the probability of drawing 2 odd cards and 2 face cards is approximately 0.079.
assuming no replacement, and order does not matter,
(a) 4! * 4/52 * 4/51 * 3/50 * 2/49 = 0.00035
(b) 4!/2!2! * 20/52 * 19/51 * 12/50 * 11/49 = 0.046