Final answer:
The probability that Isobel starts the game on her first turn is 3/11. The probability that Isobel starts the game on her second turn is 24/121. The probability that Isobel starts the game on her third turn is 192/1331. The probability that Isobel still hasn't started the game by her fourth turn is 512/1331.
Step-by-step explanation:
To find the probability that Isobel starts the game on her first turn, we need to determine the number of favorable outcomes (rolling a score of 10 or more) and divide it by the total number of possible outcomes when rolling two dice.
There are 6 different outcomes when rolling two dice:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (total 11)
Out of those 11 outcomes, there are 3 outcomes where the sum is 10 or more:
10, 11, 12
Therefore, the probability that Isobel starts the game on her first turn is 3/11.
The probability that Isobel starts the game on her second turn would be the probability of not rolling a score of 10 or more on her first turn, and then rolling a score of 10 or more on her second turn.
Since the outcomes are independent, the probability of not rolling a score of 10 or more on the first turn is 8/11 (1 - 3/11).
The probability of rolling a score of 10 or more on the second turn is the same as the probability of starting the game on the first turn, which is 3/11.
Therefore, the probability that Isobel starts the game on her second turn is (8/11) * (3/11) = 24/121.
The probability that Isobel starts the game on her third turn would be the probability of not rolling a score of 10 or more on the first and second turns, and then rolling a score of 10 or more on her third turn.
Using the same logic as before, the probability of not rolling a score of 10 or more on the first and second turns is (8/11) * (8/11) = 64/121.
The probability of rolling a score of 10 or more on the third turn is the same as the probability of starting the game on the first turn, which is 3/11.
Therefore, the probability that Isobel starts the game on her third turn is (64/121) * (3/11) = 192/1331.
To find the probability that Isobel still hasn't started the game by her fourth turn, we need to calculate the probability of not rolling a score of 10 or more in the first three turns.
The probability of not rolling a score of 10 or more on each turn is (8/11), so the probability of not rolling a score of 10 or more in three turns is (8/11)^3 = 512/1331.
The probability that Isobel still hasn't started the game by her fourth turn is 512/1331.