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

Answer:

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.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

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