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.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
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 [tex]p = 0.7[/tex]
10 sprinklers.
This means that [tex]n = 10[/tex]
What is the probability that all of the sprinklers will operate correctly in a fire?
This is P(X = 10).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 10) = C_{10,10}.(0.7)^{10}.(0.3)^{0} = 0.0282[/tex]
2.82% probability that all of the sprinklers will operate correctly in a fire
The probability that all sprinklers will operate correctly in a fire is 0.028
The given parameters are:
p = 0.7 ---- the probability that a sprinkler to activate correctly
n = 10 --- the number of sprinklers
The probability that all sprinklers activate correctly is calculated as:
[tex]P(10) = p^n[/tex]
So, we have:
[tex]P(10) = 0.7^{10[/tex]
Evaluate
[tex]P(10) = 0.028[/tex]
Hence, the probability that all sprinklers will operate correctly in a fire is 0.028
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