Question

Heather and Isabelle play a two-person game which is won by the first person to accumulate 10 points. At each turn Heather gains a point with probability of $\frac{2}{3}$ . If she doesn't get a point then Isabelle gets a point. Isabelle is now ahead 9 to 6. What is the probability that Isabelle will win? Express your answer as a common fraction.

Answer 1

Heather winning, so the probability that Isabelle will win is : 1 whole right ?

16/81 = 65 over 81

Answer 2

`(65)/(81)`

Explanation:

Observe that one person has to win the game. Then, the probability of Isabelle to win is
`1-H`, where H is the probability of Heather to win.

We will compute the probability of Heather to win:

• Each turn a player gains a point.
• Isabelle has 9 points. Then, she just need to win one more point to win the game.
• Then, the only posibility of Heather to win the game is gaining a point for the next 4 turns. The probability for this to happen is

`H=((2)/(3))^4=(16)/(81)`

Finally, as stated before, the probability of Isabelle to win is

`1-H=(65)/(81)`