Mathematics: 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?

6 is in the thousandths placeStep-by-step explanation:...

Alice and bob take turns in rolling a fair dice.

Ответ:

zaire8

15.01.2022

Mathematics

Probability of Alice win the game is $\frac{6}{11}$

Step-by-step explanation:

When a die is rolled then probability of getting 6 is $=\frac{1}{6}$

So probability of winning the game $P(W)=\frac{1}{6}$

Now probability of losing the game $P(L)=1-\frac{1}{6}=\frac{5}{6}$

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 $P(w)=\frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times\frac{5}{6} \times \frac{5}{6}\times \frac{1}{6}......$

Now taking $\frac{1}{6}$ common and applying sum of infinite GP $P(W)=\frac{1}{6}\times \frac{1}{1-\frac{25}{36}}=\frac{1}{6}\times \frac{36}{11}=\frac{6}{11}$

Probability of Alice win the game is $\frac{6}{11}$

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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?

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