Answer:
Probability of Alice win the game is [tex]\frac{6}{11}[/tex]
Step-by-step explanation:
When a die is rolled then probability of getting 6 is [tex]=\frac{1}{6}[/tex]
So probability of winning the game [tex]P(W)=\frac{1}{6}[/tex]
Now probability of losing the game [tex]P(L)=1-\frac{1}{6}=\frac{5}{6}[/tex]
It is given that Alice starts the game
So either Alice win the game in 1st attempt or 3rd or 5th .....
So probability that Alice win the game [tex]=P(W)+P(L)P(L)P(W)+P(L)P(L)P(L)P(L)P(W)+.....[/tex]
So [tex]P(w)=\frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times\frac{5}{6} \times \frac{5}{6}\times \frac{1}{6}......[/tex]
Now taking [tex]\frac{1}{6}[/tex] common and applying sum of infinite GP [tex]P(W)=\frac{1}{6}\times \frac{1}{1-\frac{25}{36}}=\frac{1}{6}\times \frac{36}{11}=\frac{6}{11}[/tex]
Probability of Alice win the game is [tex]\frac{6}{11}[/tex]
