2. Six friends all throw their coats on the bed in the spare room when they show up for Teesha's birthday party. At the end of the party they all enter the dark room and randomly grab a coat and put it on without knowing if it is actually their coat. What is the probability that exactly three of the friends got the right coat? Express your answer as a common fraction.
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r = right coat , w = wrong coat ... p(r) = 1/6 , p(w) = 5/6
(w + r)^6 = w^6 + 6 w^5 r + 15 w^4 r^2 + 20 w^3 r^3 + ...
p(3 right) = 20 w^3 r^3 = 20 * (5/6)^3 * (1/6)^3 = 625 / 11664
To find the probability that exactly three out of the six friends got the right coat, we need to calculate the number of favorable outcomes (where exactly three friends choose the right coat) and divide it by the total number of possible outcomes.
Let's analyze this step by step:
Step 1: Determine the number of ways to choose exactly three coats correctly:
Out of six friends, we can choose three friends to get their right coats in (6 choose 3) = 20 ways.
Step 2: Determine the number of ways to choose the remaining three coats incorrectly:
For each of the three friends who got their right coats, there are three remaining coats that they could potentially choose incorrectly. So, the number of ways to choose the remaining three coats incorrectly is (3^3) = 27.
Step 3: Calculate the total number of possible outcomes:
Each friend can choose one of the six coats available, resulting in a total of (6^6) = 46656 possible outcomes.
Step 4: Calculate the probability:
The probability of exactly three friends getting the right coat is given by:
(Number of favorable outcomes) / (Total number of possible outcomes)
= (20 * 27) / 46656
= 540 / 46656
Therefore, the probability that exactly three of the friends got the right coat is 540/46656, which can be simplified to 5/432.
To find the probability that exactly three friends got the right coat, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Let's analyze the problem step by step:
Step 1: Total Number of Outcomes
When the friends randomly grab a coat from the bed, there are 6 coats to choose from for the first friend, 5 coats for the second friend, 4 coats for the third friend, and so on until the sixth friend.
So, the total number of outcomes, in this case, is 6 x 5 x 4 x 3 x 2 x 1 = 720.
Step 2: Number of Favorable Outcomes
To compute the number of favorable outcomes, we need to calculate the number of ways in which exactly three of the friends will choose their own coats.
First, choose three friends out of six who will get their own coats. This can be done in C(6,3) = 6! / (3! * (6-3)!) = 6! / (3! * 3!) = 20 ways.
For these three friends, there is only one correct coat for each of them, so there is only one way for each of them to select their own coat.
For the remaining three friends, there are two possibilities for each of them: either they choose their own coat or someone else's.
So, the number of favorable outcomes is 1^3 * 2^3 = 1 * 1 * 1 * 2 * 2 * 2 = 8.
Step 3: Probability Calculation
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Number of Favorable Outcomes / Total Number of Outcomes
= 8 / 720
= 1 / 90
Therefore, the probability that exactly three friends got the right coat is 1/90.
Note: If we assume that no two friends have the same coat size, then the assumption of "randomly grabbing a coat" becomes redundant since each friend can only select their own coat. In that case, the probability would be 1.