Question

An article in Fire Technology, 2014 (50.3) studied the effectiveness of sprinklers in fire control by the number of sprinklers that activate correctly. The researchers estimate the probability of a sprinkler to activate correctly to be 0.7. Suppose that you are an inspector hired to write a safety report for a large ballroom with 10 sprinklers. Assume the sprinklers activate correctly or not independently.
What is the probability that all of the sprinklers will operate correctly in a fire?

Answer 1

Answer:

2.82% probability that all of the sprinklers will operate correctly in a fire

Step-by-step explanation:

For each sprinkler, there are only two possible outcomes. Either they will operate correctly, or they will not. The probability of a sprinkler operating correctly is independent of other sprinklers. 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.

The researchers estimate the probability of a sprinkler to activate correctly to be 0.7.

This means that $p = 0.7$

10 sprinklers.

This means that $n = 10$

What is the probability that all of the sprinklers will operate correctly in a fire?

This is P(X = 10).

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 10) = C_{10,10}.(0.7)^{10}.(0.3)^{0} = 0.0282$

2.82% probability that all of the sprinklers will operate correctly in a fire