Final answer:
To find the probability that John will roll more than 4 times until the sum of the dice rolls to 8 three times, we need to calculate the probability of rolling 8 three times or more in 4 or fewer rolls.
Step-by-step explanation:
To find the probability that John will roll more than 4 times until the sum of the dice rolls to 8 three times, we need to calculate the probability of rolling 8 three times or more in 4 or fewer rolls. Let's break it down:
First, calculate the probability of rolling a sum of 8 in one roll. There are 5 combinations that can result in a sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).
Next, calculate the probability of not rolling a sum of 8 in one roll. Since there are 36 possible outcomes when rolling two dice, and we already determined that there are 5 combinations that result in a sum of 8, there are 31 combinations that do not result in a sum of 8.
Now, we can calculate the probability of rolling a sum of 8 three times or more in 4 or fewer rolls. We can use the binomial probability formula, where n is the number of rolls (4), x is the number of successes (3), and p is the probability of success (calculated in step 1). Plugging in the values, we get:
P(X ≥ 3) = C(4, 3) * (31/36)^1 * (5/36)^3 + C(4, 4) * (31/36)^0 * (5/36)^4
Simplifying and calculating this, we find that the probability of rolling more than 4 times until the dice roll sums to 8 three times is approximately 0.0256, or 2.56%.