Two twin brothers each have two favorite sweat shirts that they wear every other day. One day, dad decides that all four sweat shirts need to be washed! He takes all four shirts and washes and dries each. After washing the shirts, he cannot tell which sweat shirts go to which brother. If dad randomly gives each brother two of the four shirts, what is the probability that each brother will get his correct two shirts?

To solve this problem, we need to understand the total number of possible outcomes and the number of favorable outcomes.

Let's start by calculating the total number of outcomes. Since there are four sweatshirts to distribute among the two brothers, each brother can potentially receive any combination of two shirts. So, the total number of outcomes is the number of ways to choose 2 shirts out of 4 for the first brother multiplied by the number of ways to choose 2 shirts out of the remaining 2 for the second brother.

The number of ways to choose 2 shirts out of 4 can be calculated using the combination formula:

C(n, r) = n! / ((n-r)! * r!)

In this case, n = 4 (the number of shirts) and r = 2 (the number of shirts each brother should receive).

For the first brother, there are C(4, 2) = 4! / ((4-2)! * 2!) = 6 ways to choose 2 shirts out of 4.

Once the first brother has received his two shirts, there are only two remaining shirts left for the second brother, so there is only one way to distribute them.

The total number of outcomes is the product of these two numbers: 6 * 1 = 6.

Now let's calculate the number of favorable outcomes, which is the number of ways both twins can get their correct two shirts.

Since we don't know which shirts go to which brother after washing, we can consider all possible assignments and count the favorable outcomes.

One way to do this is to list all the possible combinations and see how many of them satisfy the condition.

Let's assume the shirts are labeled as A, B, C, and D. Here are all the possible combinations:

1. A and B go to the first brother, C and D go to the second brother.
2. A and C go to the first brother, B and D go to the second brother.
3. A and D go to the first brother, B and C go to the second brother.
4. B and C go to the first brother, A and D go to the second brother.
5. B and D go to the first brother, A and C go to the second brother.
6. C and D go to the first brother, A and B go to the second brother.

Out of these six combinations, we can see that only in the first combination do both brothers get their correct two shirts.

Therefore, the number of favorable outcomes is 1.

Finally, we divide the number of favorable outcomes by the total number of outcomes to find the probability:

Probability = Favorable outcomes / Total outcomes = 1 / 6 = 1/6.

So, the probability that each brother will get his correct two shirts is 1/6.

There are 4 shirts A,A,B,B

There are 4 possible outcomes when drawing two shirts

AA
AB
BA
BB

So, if he draws two shirts and gives them to A, then AA is the only correct outcome, meaning p = 0.25