There are 4 white socks and 4 black socks. You randomly divides socks into pairs. What is the probability that each pair consists of 2 socks of the same colour? Express answer as a fraction.
I'm not really sure how to start this question.
well, using brute force, just list the ways you could pair up the socks.
How many of those ways fit the requirements?
There are a total of 8 socks: 4 white and 4 black. Two are randomly selected. Therefore, 8C2 or C(8,2) = 28
Out of 4 white socks, 2 white socks can be selected in 4C2 ways or C(4,2) = 6.
Out of 4 black socks, 2 black socks can be selected in 4C2 or C(4,2) = 6.
Total number of ways of selecting the same color: 6 + 6 = 12.
Probability of getting both socks of same color = 12/28 = 3/7
Alternatively, you can say there are a total of 8 socks of which 4/8 are white and 4/8 are black. To find the same color:
White socks: (4/8)*(3/7) = 3/14
Black socks: (4/8)*(3/7) = 3/14
Then, add the two probabilities: 3/14 + 3/14 = 6/14
Reduce 6/14 = 3/7 which is the same answer as above.
To solve this question, let's break it down step by step.
Step 1: Determine the total number of ways to divide the socks into pairs.
When dividing 8 socks into pairs, there are two ways to approach it: (1) Choose two socks at a time until all socks are paired, or (2) Divide the total number of socks by the number of socks per pair.
In this case, we have 8 socks to divide, and each pair consists of 2 socks. Therefore, we can use method (2) and divide 8 by 2:
Total number of ways to divide the socks into pairs = 8/2 = 4.
Step 2: Determine the number of ways to pair the socks so that each pair consists of 2 socks of the same color.
To calculate this, we will count the number of ways to pair the white socks and the number of ways to pair the black socks separately.
For the white socks:
Since there are 4 white socks, we can calculate the number of ways to pair them by using method (1) described earlier. We choose two socks at a time until all white socks are paired. The formula to calculate the number of ways to choose r objects out of n objects is given by the combination formula: C(n, r) = n! / (r!(n-r)!).
In this case, we have n = 4 white socks, and we want to choose r = 2 at a time:
Number of ways to pair the white socks = C(4, 2) = 4! / (2!(4-2)!) = 6.
For the black socks:
We follow the same calculation method as for the white socks because there are also 4 black socks:
Number of ways to pair the black socks = C(4, 2) = 4! / (2!(4-2)!) = 6.
Step 3: Calculate the probability.
The probability of the desired outcome is equal to the number of ways to pair the socks so that each pair consists of 2 socks of the same color divided by the total number of ways to divide the socks into pairs:
Probability = (Number of ways to pair same-color socks) / (Total number of ways to divide the socks into pairs)
Probability = (6 * 6) / 4 = 36 / 4 = 9.
Therefore, the probability that each pair consists of 2 socks of the same color is 9/1, which can also be simplified to 9.