No one likes me
In a school, there are 1000 boys and 1000 girls. Each girl likes exactly one boy and each boy likes exactly one girl. A person can be liked by more than one person.
If we choose a random student from the school, what is the probability that no one likes them, rounded to the nearest thousandth?
Bonus: Generalize for all values of n boys and n girls.
The answer is 0.368.
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1 solution
In general the desired probability is ( n n − 1 ) n = ( 1 − n 1 ) n , which goes to e 1 as n → ∞ .
(Note that e x = n → ∞ lim ( 1 + n x ) n . In the limit mentioned above we have x = − 1 ).
If we pick a random person (let's say it's a boy, it doesn't matter), then the probability that any one girl doesn't like him is 999/1000. Then the probability that 1000 girls don't like him is 999/1000 * 999/1000 * 999/1000. . . 1000 times = (999/1000)^1000 = ~0.368.