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## A couple decides to have $$5$$ children. Each child is equally likely to be a boy or a girl. What is the probability

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### A couple decides to have $$5$$ children. Each child is equally likely to be a boy or a girl. What is the probability

by Gmat_mission » Fri Jan 28, 2022 12:14 pm

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## Global Stats

A couple decides to have $$5$$ children. Each child is equally likely to be a boy or a girl. What is the probability that the couple has exactly $$3$$ girls and $$2$$ boys?

A. $$\dfrac1{32}$$

B. $$\dfrac1{16}$$

C. $$\dfrac5{32}$$

D. $$\dfrac5{16}$$

E. $$\dfrac16$$

Source: e-GMAT

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### Re: A couple decides to have $$5$$ children. Each child is equally likely to be a boy or a girl. What is the probability

by swerve » Fri Jan 28, 2022 12:27 pm
Gmat_mission wrote:
Fri Jan 28, 2022 12:14 pm
A couple decides to have $$5$$ children. Each child is equally likely to be a boy or a girl. What is the probability that the couple has exactly $$3$$ girls and $$2$$ boys?

A. $$\dfrac1{32}$$

B. $$\dfrac1{16}$$

C. $$\dfrac5{32}$$

D. $$\dfrac5{16}$$

E. $$\dfrac16$$

Source: e-GMAT
Number of possibilities is $$2,$$ it's either a boy or a girl.

It is mentioned that $$3$$ girls and $$2$$ boys should be the set.

When the firstborn is a $$B: B*B*G*G*G$$ (This can be done in $$4C3$$ ways by rearranging the $$B$$ and $$G$$) $$=\dfrac{1}{2}*\dfrac{1}{2}*\dfrac{1}{2}*\dfrac{1}{2}*\dfrac{1}{2}*4= 4$$ ways.

When firstborn is a $$G: G*B*B*G*G$$ (This can be done in $$4C2$$ ways) $$= \dfrac{1}{2}*\dfrac{1}{2}*\dfrac{1}{2}*\dfrac{1}{2}*\dfrac{1}{2}*6= 6$$ ways.

Adding the two scenarios$$: \dfrac{4}{32}+\dfrac{6}{32}=\dfrac{10}{32} =\dfrac{5}{16}$$

Hope this helps!

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