# Choose the Dress

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A mother bought three dress for her triplets daughters(one for each) and put the dresses in the dark. One by one the girls come and pick a dress.
What is the probability that no girl will choose her own dress?

SherlockHolmes Expert Asked on 15th October 2021 in

Let’s assume that the three dresses are indistinguishable and that each girl picks a dress at random. There are 3! = 6 ways that the girls can pick the dresses, since each girl can pick from any of the three dresses, and there are 3 ways to assign the first girl’s choice, 2 ways to assign the second girl’s choice (since one dress is already chosen), and 1 way to assign the third girl’s choice (since two dresses are already chosen).

Out of these 6 possible ways, we need to count the number of ways that no girl picks her own dress. We can use the principle of inclusion-exclusion to do this.

Let A, B, and C denote the events that the first, second, and third girl, respectively, picks her own dress. Then, we want to calculate the probability of the event not(A or B or C), which is the complement of the event A or B or C. By the principle of inclusion-exclusion, we have:

P(not(A or B or C)) = 1 – P(A or B or C) = 1 – (P(A) + P(B) + P(C) – P(A and B) – P(A and C) – P(B and C) + P(A and B and C))

Since each girl picks a dress at random, we have:

P(A) = 1/3 P(B) = 1/2 (since one dress is already chosen, the second girl has only 2 choices) P(C) = 1 (since there is only one dress left for the third girl)

Also, we have:

P(A and B) = 1/3 * 1/2 = 1/6 P(A and C) = 1/3 * 1/1 = 1/3 P(B and C) = 1/2 * 1/1 = 1/2

Finally, we have:

P(A and B and C) = 1/3 * 1/2 * 1/1 = 1/6

Substituting these values into the formula above, we get:

P(not(A or B or C)) = 1 – (1/3 + 1/2 + 1 – 1/6 – 1/3 – 1/2 + 1/6) = 1/3

Therefore, the probability that no girl will choose her own dress is 1/3.

Moshe Expert Answered on 21st February 2023.

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