Question

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he hets a total of 10. If A starts the game find the probability that B wins

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ \mathrm{Total}\mathrm{possible}\mathrm{outcome}\mathrm{for}\mathrm{A}\mathrm{to}\mathrm{win}=\left\{\right(1,6),(2,5),(3,4),(4,3),(5,2),(6,1\left)\right\} \\ =6 \\ \mathrm{sample}\mathrm{space}=36 \\ \mathrm{So},\mathrm{P}(\mathrm{A}\mathrm{winning})=\frac{6}{36}=\frac{1}{6} \\ \mathrm{and}\mathrm{P}(\mathrm{A}\mathrm{not}\mathrm{winning})=1-\frac{1}{6}=\frac{5}{6} \\ \mathrm{Total}\mathrm{possible}\mathrm{outcome}\mathrm{for}\mathrm{B}\mathrm{to}\mathrm{win}=\left\{\right(4,6),(5,5),(6,4\left)\right\} \\ =3 \\ \mathrm{P}(\mathrm{B}\mathrm{winning})=\frac{3}{36}=\frac{1}{12} \\ \mathrm{and},\mathrm{P}(\mathrm{B}\mathrm{not}\mathrm{winning})=1-\frac{1}{12}=\frac{11}{12} \\ \mathrm{A}\mathrm{and}\mathrm{B}\mathrm{keep}\mathrm{playing}\mathrm{until}\mathrm{B}\mathrm{wins}\mathrm{the}\mathrm{game}; \\ \mathrm{So}, \\ \mathrm{Probability}\mathrm{of}\mathrm{B}\mathrm{winning}; \\ (\mathrm{B}\mathrm{wins}\mathrm{in}1\mathrm{st}\mathrm{throw})+(\mathrm{B}\mathrm{wins}\mathrm{in}2\mathrm{nd}\mathrm{throw})+(\mathrm{B}\mathrm{wins}\mathrm{in}3\mathrm{rd}\mathrm{throw})+....... \\ (\frac{5}{6}\times \frac{1}{12})+(\frac{5}{6}\times \frac{11}{12}\times \frac{5}{6}\times \frac{1}{12})+(\frac{5}{6}\times \frac{11}{12}\times \frac{5}{6}\times \frac{11}{12}\times \frac{5}{6}\times \frac{1}{12})+........\mathrm{upto}\mathrm{infinity} \\ =\frac{5}{72}\left[1+\frac{55}{72}+\frac{3025}{5184}+.........\right] \\ \mathrm{Now},1+\frac{55}{72}+\frac{3025}{5184}+......... \\ \mathrm{It}\mathrm{is}\mathrm{Geometric}\mathrm{Progression}\mathrm{Series};\mathrm{with}\mathrm{a}=1\mathrm{and}\mathrm{r}=\frac{55}{72} \\ \mathrm{Sum}=\frac{\mathrm{a}}{1-\mathrm{r}} \\ =\frac{1}{1-{\displaystyle \frac{55}{72}}} \\ =\frac{72}{17} \\ \mathbf{Therefore}\mathbf{,}\mathbf{}\mathbf{probability}\mathbf{}\mathbf{of}\mathbf{}\mathbf{B}\mathbf{}\mathbf{winning}\mathbf{}\mathbf{the}\mathbf{}\mathbf{game}\mathbf{=}\frac{\mathbf{5}}{\mathbf{72}}\mathbf{\times }\frac{\mathbf{72}}{\mathbf{17}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{5}}{\mathbf{17}} \\ \mathrm{Regards}. \end{gathered}$$