Question

Alice and Bob take turns in rolling a fair dice. Whoever gets "6" first wins the game. Alice starts the game. What are the chances that Alice wins?

Answer 1

a, c, and e

Explanation:

I did ed. :)

Answer 2

Probability of Alice win the game is
`(6)/(11)`

Explanation:

When a die is rolled then probability of getting 6 is
`=(1)/(6)`

So probability of winning the game
`P(W)=(1)/(6)`

Now probability of losing the game
`P(L)=1-(1)/(6)=(5)/(6)`

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
`=P(W)+P(L)P(L)P(W)+P(L)P(L)P(L)P(L)P(W)+.....`

So
`P(w)=(1)/(6)+(5)/(6)* (5)/(6)* (1)/(6)+(5)/(6)* (5)/(6)*(5)/(6) * (5)/(6)* (1)/(6)......`

Now taking
`(1)/(6)` common and applying sum of infinite GP
`P(W)=(1)/(6)* (1)/(1-(25)/(36))=(1)/(6)* (36)/(11)=(6)/(11)`

Probability of Alice win the game is
`(6)/(11)`