Question

2 girls have 25 outfits each. Each girl has one and only one outfit that matches the other girls. What is the probability that the girls wear matching outfits twice in 21 days?

Answer 1

m1 = for first girl the number of ways to wear the outfit for each day= 25

m2 = for second girl number of ways to wear the outfit for each day = 25

First girl wearing the outfit and second girl wearing the outfit are independent

Hence total number of ways to wear the outfit for each day = 25 * 25 = 625

Number of ways when two girls wear the matching output = 1

p =P ( Two girls wear matching outfits for each day ) = 1 / 625

n = number of days = 21

X : Number of days in which two girls wearing matching outfit

X take values 0,1,...........,21.

The random variable X follows binomial distribution with n = 21 and p = 1 / 625

The probability mass function is given by

P ( Girls wear matching outfit twice in 21 days) = P(X=2)

= 0.00005

Alternative solution.

Since p is small and np = 21 * 1/625 = 0.0016 (finite)

as p is small and np is finite the probability distribution of X tends Poisson with parameter np = lambda.

The probability mass function of X is

P ( Girls wear matching outfit twice in 21 days) = P(X=2)

= 0