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Sagot :
Answer:
0.9606 = 96.06% probability that the system will function tomorrow
Step-by-step explanation:
For each component, there are only two possible outcomes. Either it works, or it does not. Components are independent. This means that 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.
Probability of a component working:
0.7 of 0.2(rain)
0.9 of 1 - 0.2 = 0.8(no rain). So
[tex]p = 0.7*0.2 + 0.9*0.8 = 0.86[/tex]
0.86 - 86% probability that the system will function tomorrow
6 components:
This means that [tex]n = 6[/tex]
What is the probability that the system will function tomorrow
This is
[tex]P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 4) = C_{6,4}.(0.86)^{4}.(0.14)^{2} = 0.1608[/tex]
[tex]P(X = 5) = C_{6,5}.(0.86)^{5}.(0.14)^{1} = 0.3952[/tex]
[tex]P(X = 6) = C_{6,6}.(0.86)^{6}.(0.14)^{0} = 0.4046[/tex]
[tex]P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) = 0.1608 + 0.3952 + 0.4046 = 0.9606[/tex]
0.9606 = 96.06% probability that the system will function tomorrow
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