Question

Two players A and B plays a series of ‘2n’ games . Each game can result in either a win or loss for A. Find the total number of ways in which A can win the series of these games . ( all the games are to be played).

Answer 1

$$\begin{gathered} \mathrm{Hi}, \\ \mathrm{The}\mathrm{winning}\mathrm{probbility}\mathrm{for}\mathrm{both}\mathrm{are}1/2\mathrm{so} \\ \mathrm{For}\mathrm{A}\mathrm{to}\mathrm{win} \\ =\frac{1}{2}+\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}+\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}+......2\mathrm{n}\mathrm{times} \\ =\frac{{\displaystyle \frac{1}{2}}\left[1-{\left(\frac{1}{4}\right)}^{2\mathrm{n}}\right]}{1-{\displaystyle \frac{1}{4}}} \\ =\frac{2}{3}\left[1-{\left(\frac{1}{4}\right)}^{2\mathrm{n}}\right] \\ \mathrm{Regards} \end{gathered}$$