Question

What is the probability of mission success if there are 16 riverine patrol boats scheduled for a 20-hour mission and at least 10 patrol boats must operate for the entire duration of the mission if each boat has a failure rate of 1 failure per 100 hours?

A. 99.5%
B. 95.0%
C. 90.0%
D. 85.5%

Answer 1

The probability of mission success, given the stated conditions, is approximately 99.5%. So, the answer is A. 99.5%.

Here's how we can calculate the probability of mission success:

Calculate the probability of a single boat failing during the mission:

Failure rate per hour = 1/100

Probability of failure in 20 hours = 1 - (1 - 1/100)^20 ≈ 0.1813 (using binomial approximation)

Calculate the probability of not enough boats failing (at least 10 must stay operational):

We need 10 or more boats to not fail: P(success) = P(10+ boats operational) + P(11+ boats operational) + ... + P(16 boats operational)

This can be calculated using the binomial probability distribution formula for each scenario and summing them up. However, considering the low failure rate and large number of boats, it's easier to approximate using the complementary probability:

P(failure) = P(9 or fewer boats operational) = 1 - P(10+ boats operational)

P(failure) ≈ 16 * 0.1813^7 ≈ 0.0049 (using binomial approximation)

P(success) ≈ 1 - P(failure) ≈ 1 - 0.0049 ≈ 0.9951

Convert the probability to a percentage:

P(success) ≈ 0.9951 * 100% ≈ 99.5%

Therefore, the probability of mission success, given the stated conditions, is approximately 99.5%.

So, the answer is A. 99.5%.