Final answer:
The expected number of rolls until a 6 is rolled, given that you don't roll a 6 on the first two rolls, is 7.
Step-by-step explanation:
To find the expected number of rolls until a 6 is rolled given that you don't roll a 6 on the first two rolls, we can use the concept of geometric distribution.
The geometric distribution models the number of trials needed to achieve a success in a sequence of independent trials, where each trial has the same probability of success, in this case, rolling a 6. The probability of rolling a 6 on any given roll is 1/6, so the probability of not rolling a 6 on any given roll is 5/6.
Since we know that the first two rolls do not result in a 6, we essentially start the geometric distribution from the third roll. So, the expected number of rolls until a 6 is rolled can be calculated as:
E(X) = 1 + (1/p)
where p = 1/6.
Substituting the value of p, we get:
E(X) = 1 + (1/(1/6)) = 1 + 6 = 7
Therefore, the expected number of rolls until a 6 is rolled, given that you don't roll a 6 on the first two rolls, is 7.