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Question Number 160314 by MathsFan last updated on 27/Nov/21
suppose the probability of a child  being a boy is 0.5. Find the   probability that a family of    3 children will have   (i) at least two boys  (ii) exactly two boys  (iii) all girls
$${suppose}\:{the}\:{probability}\:{of}\:{a}\:{child} \\ $$$${being}\:{a}\:{boy}\:{is}\:\mathrm{0}.\mathrm{5}.\:{Find}\:{the}\: \\ $$$${probability}\:{that}\:{a}\:{family}\:{of}\: \\ $$$$\:\mathrm{3}\:{children}\:{will}\:{have}\: \\ $$$$\left({i}\right)\:{at}\:{least}\:{two}\:{boys} \\ $$$$\left({ii}\right)\:{exactly}\:{two}\:{boys} \\ $$$$\left({iii}\right)\:{all}\:{girls} \\ $$
Commented by yeti123 last updated on 27/Nov/21
using binomial distribution:  P(x) =  ((3),(x) )(0.5^x )(0.5^(3−x) )  (i) P(x ≥ 2) = P(2) + P(3)                             =  ((3),(2) )(0.5^2 )(0.5) +  ((3),(3) )(0.5^3 )  (ii) P(2) =  ((3),(2) )(0.5^2 )(0.5)  (iii) P(0) =  ((3),(0) )(1)(0.5^3 )
$$\mathrm{using}\:\mathrm{binomial}\:\mathrm{distribution}: \\ $$$$\mathrm{P}\left({x}\right)\:=\:\begin{pmatrix}{\mathrm{3}}\\{{x}}\end{pmatrix}\left(\mathrm{0}.\mathrm{5}^{{x}} \right)\left(\mathrm{0}.\mathrm{5}^{\mathrm{3}−{x}} \right) \\ $$$$\left({i}\right)\:\mathrm{P}\left({x}\:\geqslant\:\mathrm{2}\right)\:=\:\mathrm{P}\left(\mathrm{2}\right)\:+\:\mathrm{P}\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\begin{pmatrix}{\mathrm{3}}\\{\mathrm{2}}\end{pmatrix}\left(\mathrm{0}.\mathrm{5}^{\mathrm{2}} \right)\left(\mathrm{0}.\mathrm{5}\right)\:+\:\begin{pmatrix}{\mathrm{3}}\\{\mathrm{3}}\end{pmatrix}\left(\mathrm{0}.\mathrm{5}^{\mathrm{3}} \right) \\ $$$$\left({ii}\right)\:\mathrm{P}\left(\mathrm{2}\right)\:=\:\begin{pmatrix}{\mathrm{3}}\\{\mathrm{2}}\end{pmatrix}\left(\mathrm{0}.\mathrm{5}^{\mathrm{2}} \right)\left(\mathrm{0}.\mathrm{5}\right) \\ $$$$\left({iii}\right)\:\mathrm{P}\left(\mathrm{0}\right)\:=\:\begin{pmatrix}{\mathrm{3}}\\{\mathrm{0}}\end{pmatrix}\left(\mathrm{1}\right)\left(\mathrm{0}.\mathrm{5}^{\mathrm{3}} \right) \\ $$
Commented by MathsFan last updated on 27/Nov/21
thank you sir
$${thank}\:{you}\:{sir} \\ $$

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