Question

suppose the lifetime (in months) of an engine, working under hazardous conditions, has an exponential distribution with mean of 5 months. an engine is termed successful if its lifetime exceeds 10 months. in a sample of 10 engines, what is the probability of at least 2 successful engines?

Answer 1

To find the probability of at least 2 successful engines in a sample of 10 engines, we can use the binomial distribution and the probability that an engine's lifetime exceeds 10 months. The probability of at least 2 successful engines is 1 minus the probability of 0 or 1 successful engines.

Step-by-step explanation:

To find the probability of at least 2 successful engines out of a sample of 10 engines, we can use the binomial distribution. Let's define the random variable X as the number of successful engines in the sample.

The probability of at least 2 successful engines is equal to 1 minus the probability of 0 or 1 successful engines. We can calculate this probability using the binomial probability formula: P(X ≥ 2) = 1 - P(X = 0) - P(X = 1).

For each engine, the probability of it being successful is the probability that its lifetime exceeds 10 months. Since the lifetime of an engine follows an exponential distribution with a mean of 5 months, the probability of a single engine being successful is given by:

P(Success) = P(Engine lifetime > 10) = 1 - P(Engine lifetime ≤ 10) = 1 - e(-10/5) = 1 - e(-2) ≈ 0.8647

Using this value, we can now calculate the probability of at least 2 successful engines in a sample of 10 using the binomial probability formula.