Question Number 123390 by aurpeyz last updated on 25/Nov/20
$${in}\:{gambling}\:{game}\:{a}\:{man}\:{is}\:{paid}\:\$\mathrm{5} \\ $$$${if}\:{he}\:{gets}\:{all}\:{heads}\:{or}\:{all}\:{tails}\:{when} \\ $$$$\mathrm{3}\:{coins}\:{are}\:{tossed}\:{and}\:{he}\:{pays}\:{out} \\ $$$$\$\mathrm{3}\:{if}\:{either}\:\mathrm{1}\:{or}\:\mathrm{2}\:{heads}\:{appears}. \\ $$$${calculate}\:{his}\:{expected}\:{gain} \\ $$
Answered by talminator2856791 last updated on 25/Nov/20
$$\: \\ $$$$\:{P}\left(\$\mathrm{3}\right)\:=\:\frac{\mathrm{6}}{\mathrm{8}}\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\:{P}\left(\$\mathrm{5}\right)=\:\frac{\mathrm{2}}{\mathrm{8}}\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{3}}\:{P}\:\left(\$\mathrm{3}\right)\:=\:{P}\:\left(\$\mathrm{5}\right) \\ $$$$\:\mathrm{he}\:\mathrm{gains}\:\$\mathrm{5}\:\mathrm{he}\:\mathrm{loses}\:\$\mathrm{9} \\ $$$$\:\mathrm{expected}\:\mathrm{gain}\:\$−\mathrm{4} \\ $$