A standard six-sided die is rolled repeatedly until either a six comes up or two fives come up in a row. Find the probability that the die is rolled at least four times.

To find the probability that the die is rolled at least four times, we need to find the probability that a six or two fives in a row do not come up in the first three rolls.

The probability of not getting a six in one roll is 5/6. So, the probability of not getting a six in three rolls is (5/6)^3 = 125/216.

The probability of not getting two fives in a row in three rolls is a bit more complicated. There are 6^3 = 216 possible outcomes in three rolls. The outcomes that include two fives in a row are: 55X, X55, and 555, where X can be any number from 1 to 6. There are 6 outcomes for 55X, 6 outcomes for X

The probability that the die is rolled at least four times means that neither a six comes up nor two fives come up in a row in the first three rolls.

The probability that a six comes up in a single roll is 1/6, so the probability that a six does not come up is 5/6.

The probability that two fives come up in a row in two rolls is (1/6) * (1/6) = 1/36, so the probability that two fives do not come up in a row is 1 - 1/36 = 35/36.

For the die to be rolled at least four times, neither a six nor two fives in a row can come up in the first three rolls. The events of each roll are independent, so we can multiply the probabilities together:

P(at least four rolls) = P(no six in first roll) * P(no six in second roll) * P(no six or two fives in a row in third roll) = (5/6) * (5/6) * [(5/6) * (35/36)] = 125/216 * 35/36 = 4375/7776 = 0.5625

So, the probability that the die is rolled at least four times is approximately 0.5625.