Question

A die is continuously rolled until the total sum of all rolls exceeds 175 What is the probability that at least 45 rolls are necessary?

Answer 1

To find the probability of at least 45 rolls being necessary to exceed a sum of 175 when rolling a die, we can use a geometric series. The probability is given by the formula 6/(5 * 6^45).

Step-by-step explanation:

To find the probability that at least 45 rolls are necessary, we need to consider the cumulative probability of rolling a die until the sum exceeds 175. Let's break it down step by step:

1. The probability of rolling a sum of 1 on the first roll is 1/6.
2. The probability of rolling a sum of 2 on the second roll is (1/6) * (1/6) = 1/36.
3. We continue this pattern, multiplying the probabilities each time.
4. The probability of rolling a sum of 175 on the 45th roll is (1/6)^(44) * (1/6) = 1/6^45.

To find the probability of at least 45 rolls, we sum all the probabilities from the 45th roll to infinity. This can be represented as a geometric series:

P(at least 45 rolls) = (1/6^45) + (1/6^46) + (1/6^47) + ...

The sum of this geometric series is given by the formula:

P(at least 45 rolls) = (1/6^45) * (1 / (1 - 1/6)) = (1/6^45) * (6/5) = 6/5 * (1/6^45) = 6/(5 * 6^45).