In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers (1, 2, 3, 4, 5, 6, and 7). Among all the cards of each color, there is exactly one card labeled with each number. The cards in Ms. Q's deck are shown below.
Yunseol draws 5 cards from Ms. Q's deck. What is the probability that three cards have the same number?
The probability that three cards have the same number is 1/35.
To find the probability that three cards have the same number out of 5 cards drawn from Ms. Q's deck, we first need to determine the total number of possible outcomes.
Total number of possible outcomes:
We have a total of 4 colors and 7 numbers, so the total number of cards in Ms. Q's deck is 4 * 7 = 28 cards.
From this deck, Yunseol draws 5 cards. The total number of ways to choose 5 cards from 28 cards is calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of cards and r is the number of cards drawn.
In this case, n = 28 and r = 5, so the total number of possible outcomes is:
C(28, 5) = 28! / (5!(28-5)!) = 28! / (5!23!) = (28*27*26*25*24) / (5*4*3*2*1) = 98,280
Now, we need to determine the number of favorable outcomes, i.e., the number of ways to choose 3 cards with the same number out of the 5 cards drawn.
Number of favorable outcomes:
Since there are 7 numbers in total, we have 7 options for the number of the three cards that will have the same number. Once we choose the number, we need to select 3 cards with that number from a total of 4 colors. Hence, the number of favorable outcomes is:
7 * C(4, 3) = 7 * (4! / (3!(4-3)!)) = 7 * (4! / (3!1!)) = 7 * (4*3*2*1 / (3*2*1)) = 7 * 4 = 28
Therefore, the number of favorable outcomes is 28.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes = 28 / 98,280 ≈ 0.000285
So, the probability that three cards have the same number is approximately 0.000285, or about 0.0285%.
To find the probability that three cards have the same number when drawing 5 cards from Ms. Q's deck, we need to calculate two probabilities:
1. The probability of choosing three cards with the same number
2. The probability of choosing two additional cards with different numbers than the previous three chosen cards.
Step 1: Calculate the probability of choosing three cards with the same number.
- There are four colors in total: red, green, blue, and yellow.
- For each color, there are seven numbers: 1, 2, 3, 4, 5, 6, and 7.
- We need to select three cards with the same number from one of these seven options.
- For each number, there are four different colors.
The probability of choosing three cards with the same number can be calculated as follows:
Probability = (number of ways to choose three cards with the same number) / (total number of possible combinations when drawing 5 cards)
To calculate the number of ways to choose three cards with the same number:
- We have seven options for the number.
- We choose one of these seven numbers, and there are four different colors for that number.
- We need to choose three cards out of the four colors, giving us (4 choose 3) = 4 possible combinations.
To calculate the total number of possible combinations when drawing 5 cards:
- Since each card can be one of four different colors, at each draw, there are 4 options.
- Therefore, the total number of possible combinations is (4^5) = 1024.
The number of ways to choose three cards with the same number is 4, and the total number of possible combinations is 1024.
So the probability of choosing three cards with the same number is:
Probability = 4 / 1024
Step 2: Calculate the probability of choosing two additional cards with different numbers than the previous three chosen cards.
- After selecting three cards with the same number, there are only four remaining numbers to choose from (since we cannot choose the same number as the three already chosen).
- Therefore, we have four options for each of the two remaining cards.
The probability of choosing two additional cards with different numbers is:
Probability = (number of ways to choose two different numbers) / (total number of possible combinations when drawing 2 cards)
To calculate the number of ways to choose two different numbers:
- We have four options for the first number (to ensure it is different from the three already chosen).
- We have three options for the second number (since it also needs to be different from the three already chosen).
- Therefore, the number of ways to choose two different numbers is 4 * 3 = 12.
To calculate the total number of possible combinations when drawing 2 cards:
- Since each card can be one of four different colors, at each draw, there are 4 options.
- Therefore, the total number of possible combinations is (4^2) = 16.
The number of ways to choose two different numbers is 12, and the total number of possible combinations is 16.
So the probability of choosing two additional cards with different numbers is:
Probability = 12 / 16
Step 3: Calculate the overall probability of choosing three cards with the same number.
To calculate the overall probability, we multiply the probability from Step 1 (choosing three cards with the same number) by the probability from Step 2 (choosing two additional cards with different numbers).
Overall Probability = Probability (Step 1) * Probability (Step 2)
Overall Probability = (4 / 1024) * (12 / 16)
Overall Probability = 48 / 16384
Overall Probability = 3 / 1024
Therefore, the probability that three cards have the same number when drawing 5 cards from Ms. Q's deck is 3 / 1024.