Of a group of children, 0.2 are boys and 0.8 are girls. Of the boys, 0.7 have brown eyes; of the girls, 0.4 have brown eyes. A child is selected at random from the group.

(a) Find the probability that the child is a girl.


(b) Find P(brown eyes | boy).


(c) Find the probability that the child is a boy with brown eyes.


(d) Find the probability that the child is a girl with brown eyes.
 

(e) Find the probability that the child has brown eyes.


(f) Find the probability that the child is a girl, given that the child has brown eyes. (Round your answer to three decimal places.)

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1 Answer

It makes it easier if we change to actual numbers. If there are 100 children, there are 20 boys, 80 girls, 14 brown-eyed boys, 32 brown-eyed girls, 46 brown-eyed children.

a) P(girl)=0.80 or 80%.

b) If this means, given a boy, the probability of his having brown eyes, then the probability is 0.7 or 70%. If it means, given that the child has brown eyes (46 children), then 14/46=7/23=0.304 approx is the probability that it is a boy who has brown eyes (rather than a girl).

c) P(brown-eyed boy)=0.14 or 14% (14 boys out of 100 children have brown eyes).

d) P(brown-eyed girl)=0.32 or 32%.

e) P(brown-eyed child)=0.46 or 46%.

f) Given that the child has brown eyes (46 children), 32/46=0.696 or 69.6% is the probability that it’s a girl.

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