Two kittens, at least one female

Question

A breeder has a litter of two kittens. Each kitten is independently male or female with probability $\tfrac{1}{2}$ . She tells you that at least one of the two is female.

Given this, what is the probability that both kittens are female?

Accepted Answer

Before any information, the two kittens fall into four equally likely combinations, writing the pair in birth order: MM,\quad MF,\quad FM,\quad FF,\qquad 	ext{each with probability } 	frac14. The statement "at least one is female" rules out only MM. The three survivors, MF, FM, FF, remain equally likely, so each now has probability 	frac{1}{3}. Both female is the single case FF, so P(	ext{both female} \mid 	ext{at least one female}) = \frac{1}{3}. The tempting answer 	frac{1}{2} comes from imagining "the other kitten is a coin flip," but that reasoning quietly assumes you were told abou…

Two kittens, at least one female

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