To solve this problem, we need to find the probability that Sam's first upgrade to first class occurs after the third flight.

Understanding the Problem

The probability that Sam gets upgraded on any given flight is $p = 0.10$. This is a binomial process, where each flight is an independent trial with two possible outcomes: upgrade (success) or no upgrade (failure).

We are asked to find the probability that Sam's first upgrade (the first "success") occurs after the third flight. This can be thought of as a scenario where there are no upgrades (all "failures") for the first three flights.

Formulation as a Geometric Distribution

To solve this, we use the concept of a geometric distribution. The geometric distribution describes the probability of the first success on the $n$-th trial. The probability that the first success occurs after the third flight is the same as the probability that there are no successes in the first three flights.

Step-by-Step Solution

1. Probability of not being upgraded on a single flight: $\text{Probability of no upgrade} = 1 - p = 1 - 0.10 = 0.90$

2. Probability of no upgrades in the first three flights: Since each flight is independent, the probability that Sam does not get upgraded on all three flights is: $(0.90)^3$

3. Calculate the Probability: $(0.90)^3 = 0.9 \times 0.9 \times 0.9 = 0.729$

Conclusion

The probability that Sam’s first upgrade will occur after the third flight is 0.729.

Would you like more details, or do you have any other questions?

Related Questions:

1. What is the probability that Sam gets upgraded for the first time on the fourth flight?
2. How would the probability change if the upgrade rate increased to 0.15?
3. What is the expected number of flights until Sam gets his first upgrade?
4. What is the probability that Sam will be upgraded at least once in the first five flights?
5. How do you calculate the variance of the number of flights until the first upgrade?

Tip: When dealing with probabilities involving independent events, consider whether the situation fits common distributions like binomial or geometric.