Question

Nuclear power plants have redundant components in important systems to reduce the chance of catastrophic failure. Suppose that a plant has three gauges to measure the level of the coolant in the reactor core and each gauge has a 0.02 probability of failing. What is the probability that none of the gauges fails

Answer 1

The probability that none of the gauges fails is 0.941192.

Step-by-step explanation:

To find the probability that none of the gauges fails, we need to calculate the probability that each gauge does not fail and then multiply those probabilities together.

Since each gauge has a 0.02 probability of failing, the probability that a gauge does not fail is 1 - 0.02 = 0.98

Therefore, the probability that none of the gauges fails is 0.98 x 0.98 x 0.98 = 0.941192

Answer 2

0.9412 = 94.12% probability that none of the gauges fails

Step-by-step explanation:

For each gauge, there are only two possible outcomes. Either it fails, or it does not. The probability of a gauge failing is independent of other gauges. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

3 gouges, each with a 0.02 probability of failing.

This means that
`n = 3, p = 0.02`

What is the probability that none of the gauges fails?

This is P(X = 0).

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(3,0).(0.02)^(0).(0.98)^(3) = 0.9412`

0.9412 = 94.12% probability that none of the gauges fails