There are two distributions for heights men and women, find the probability that a random person with height of 170 is woman

probability probability-distributions normal-distribution

Solution 1:

You use the frequency distributions to find the likelihood that woman is $170 cm$ and the probability that a man is $170 cm$

Then we use Bayes theorem to find the probability of a woman given the height is 170 cm

$\frac {P(170|woman)}{P(170|woman) + P(170|man)}$

There is a small shortcut here because the number of men and the number of women in the population is equal. If it were not we would need weighting factors for the sizes of the populations.

$\frac {P(170|woman)(\text{female pop})}{P(170|woman)(\text{female pop}) + P(170|man)(\text{male pop})}$

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