Three cards are chosen at random from a standard 52-card deck. What is the probability that they are not all the same color?
3 card combinations ... 52C3
black 3 card combinations ... 26C3
red 3 card combinations ... 26C3
probability not same color = 1 - (probability of same color) = 1 - [(2 * 26C3) / 52C3]
To find the probability that the three cards are not all the same color, we need to find the total number of favorable outcomes and divide it by the total number of possible outcomes.
Step 1: Find the total number of possible outcomes
A standard 52-card deck consists of 26 red cards (13 hearts and 13 diamonds) and 26 black cards (13 clubs and 13 spades). So, there are a total of C(52, 3) ways to choose 3 cards from a deck of 52.
Step 2: Find the number of favorable outcomes
To have not all the same color, we need to consider two cases:
Case 1: Two red cards and one black card.
The number of ways to choose 2 cards from 13 red cards is C(13, 2), and the number of ways to choose 1 card from 26 black cards is C(26, 1). So, the total number of favorable outcomes for this case is C(13, 2) * C(26, 1).
Case 2: Two black cards and one red card.
The number of ways to choose 2 cards from 26 black cards is C(26, 2), and the number of ways to choose 1 card from 13 red cards is C(13, 1). So, the total number of favorable outcomes for this case is C(26, 2) * C(13, 1).
Step 3: Calculate the probability
The probability is given by the ratio of the number of favorable outcomes to the number of possible outcomes:
Probability = (favorable outcomes) / (total outcomes)
= (C(13, 2) * C(26, 1) + C(26, 2) * C(13, 1)) / C(52, 3)
Step 4: Calculate the final answer
By evaluating the combinations, we can simplify the expression for the probability to:
Probability = (78 * 26 + 325 * 13) / 22100
= 2028 / 22100
≈ 0.0918
Therefore, the probability that the three cards are not all the same color is approximately 0.0918, or 9.18%.
To find the probability that three randomly chosen cards from a standard 52-card deck are not all the same color, we first need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
We have to choose three cards from a deck of 52 cards without considering the order in which they are chosen. This can be calculated using the combination formula, also known as "n choose k". In this case, the number of possible outcomes is 52 choose 3, which is calculated as:
C(52, 3) = 52! / (3! * (52 - 3)!) = 22,100.
Number of favorable outcomes:
To find the number of favorable outcomes, we need to consider that the three cards are not all the same color. There are two cases:
1. Two cards of one color and one card of the other color.
2. Three cards of mixed colors.
Case 1: Two cards of one color and one card of the other color.
There are two possible scenarios for this case:
a) Choosing two red cards and one black card.
The number of favorable outcomes for this scenario is calculated as:
C(26, 2) * C(26, 1) = 26! / (2! * (26 - 2)!) * 26! / (1! * (26 - 1)!) = 8,450.
b) Choosing two black cards and one red card.
The number of favorable outcomes for this scenario is also 8,450.
Therefore, the total number of favorable outcomes for case 1 is:
8,450 + 8,450 = 16,900.
Case 2: Three cards of mixed colors.
The number of favorable outcomes for this case is calculated as:
C(26, 3) = 26! / (3! * (26 - 3)!) = 2,600.
Adding the favorable outcomes from both cases, we get:
16,900 + 2,600 = 19,500.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(not all the same color) = 19,500 / 22,100 ≈ 0.88.
So, the probability that three cards chosen at random from a standard 52-card deck are not all the same color is approximately 0.88 or 88%.