Mathematics - Probability Question with Solution | TestHub

The man has to win atleast $4$ times.

$\therefore$ Required probability

$={}^{7}C_4{\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^3+{}^{7}C_5{\left(\frac{1}{2}\right)}^5{\left(\frac{1}{2}\right)}^2+{}^{7}C_6{\left(\frac{1}{2}\right)}^6\left(\frac{1}{2}\right)+{}^{7}C_7{\left(\frac{1}{2}\right)}^7$

$=\left({}^{7}C_4+{}^{7}C_5+{}^{7}C_6+{}^{7}C_7\right)\times \frac{1}{2^7}=\frac{64}{2^7}=\frac{1}{2}$