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ipn [44]
2 years ago
5

A k out of n system is one in which there is a group of n components, and the system will function if at least k of the componen

ts function. For a certain 4 out of 6 system, assume that on a rainy day each component has probability 0.7 of functioning, and that on a non rainy day each component has probability 0.9 of functioning.Assume that the probability of rain tomorrow is 0.20. What is the probability that the system will function tomorrow
Mathematics
1 answer:
Solnce55 [7]2 years ago
6 0

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.

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.

Probability of a component working:

0.7 of 0.2(rain)

0.9 of 1 - 0.2 = 0.8(no rain). So

p = 0.7*0.2 + 0.9*0.8 = 0.86

0.86 - 86% probability that the system will function tomorrow

6 components:

This means that n = 6

What is the probability that the system will function tomorrow

This is

P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6)

In which

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

P(X = 4) = C_{6,4}.(0.86)^{4}.(0.14)^{2} = 0.1608

P(X = 5) = C_{6,5}.(0.86)^{5}.(0.14)^{1} = 0.3952

P(X = 6) = C_{6,6}.(0.86)^{6}.(0.14)^{0} = 0.4046

P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) = 0.1608 + 0.3952 + 0.4046 = 0.9606

0.9606 = 96.06% probability that the system will function tomorrow

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