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