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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.

Respuesta :

dianejones47
Heather winning, so the probability that Isabelle will win is : 1 whole right ? 

16/81 = 65 over 81
josebarradas1607

Answer:

[tex]\frac{65}{81}[/tex]

Step-by-step explanation:

Observe that one person has to win the game. Then, the probability of Isabelle to win is [tex]1-H[/tex], 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

[tex]H=(\frac{2}{3})^4=\frac{16}{81}[/tex]

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

[tex]1-H=\frac{65}{81}[/tex]

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