Final answer:
The probability that Isabelle will win the game from a score of 9 to 6 is 1/3. This is because Heather has a 2/3 chance of winning a point each turn, and if Heather does not win a point, Isabelle does, which would end the game.
Step-by-step explanation:
To determine the probability that Isabelle will win the game from a score of 9 to 6 against Heather, we only need to consider the outcomes of the next few rounds.
Heather needs 4 points to win while Isabelle only needs one point. Since at each turn Heather gains a point with a probability of 2/3 and Isabelle obtains a point otherwise (with a probability of 1/3), we can calculate the probability that Isabelle will win in the next round directly.
The probability that Isabelle wins at the next turn is simply the probability that Heather does not gain a point, which is 1/3.
Since Isabelle only needs one point to win and this can be accomplished in the very next round, the probability of Isabelle winning the game is 1/3.
It is not necessary to consider further rounds as the game ends as soon as Isabelle wins a point.