A communication system consists of n components components. Each of these functions independently with probability p. The system function correctly if and only if at least half of its components functions. For what range of p, the probability that a five-components system functions correctly is higher than the probability that a three-components system functions correctly?a)[0.4,0.6]b)[0,0.5]c)[0,1]d)[0.5,1]Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Understanding the problem:
We are given a communication system consisting of n components, where each component functions independently with probability p. The system functions correctly only if at least half of its components function.

We need to find the range of values for p, where the probability that a five-component system functions correctly is higher than the probability that a three-component system functions correctly.

Probability of a system functioning correctly:
To find the probability that a system functions correctly, we need to consider the different possible scenarios where at least half of the components function.

For a three-component system:
In a three-component system, for the system to function correctly, at least two components must function. Let's consider the possibilities:
- All three components function: p * p * p
- Two components function, one component fails: p * p * (1-p)
- One component functions, two components fail: p * (1-p) * (1-p)
- All three components fail: (1-p) * (1-p) * (1-p)

The probability that a three-component system functions correctly is the sum of the probabilities for the first two scenarios:
P(3-components system functions) = p^3 + 3p^2(1-p)

For a five-component system:
In a five-component system, for the system to function correctly, at least three components must function. Let's consider the possibilities:
- All five components function: p * p * p * p * p
- Four components function, one component fails: p * p * p * p * (1-p)
- Three components function, two components fail: p * p * p * (1-p) * (1-p)
- All other possibilities will have less than half of the components functioning, so we don't consider them.

The probability that a five-component system functions correctly is the sum of the probabilities for the first three scenarios:
P(5-components system functions) = p^5 + 5p^4(1-p) + 10p^3(1-p)^2

Comparing the probabilities:
To find the range of values for p where the probability of a five-component system functioning correctly is higher than a three-component system, we need to compare the two probabilities.

P(5-components system functions) > P(3-components system functions)

By substituting the expressions we derived for the probabilities, we get:
p^5 + 5p^4(1-p) + 10p^3(1-p)^2 > p^3 + 3p^2(1-p)

Simplifying the inequality gives:
p^5 + 5p^4 - 5p^5 + 10p^3 - 20p^4 + 10p^5 > p^3 + 3p^2 - 3p^3

Rearranging the terms and simplifying further gives:
3p^5 - 15p^4 + 10p^3 - 3p^2 + 3p > 0

Factoring out p gives:
p(3p^4 - 15p^3 + 10p^2 - 3p + 3) > 0

Range of p:
To find the range of p where the inequality holds true, we need

Answer

Five component system functions correctly if at least 3 components function and 3 component system component function correctly if at least two components function.

Similar Mathematics Doubts

An infinite sequence of independent trials is to bo performed. Each Trial results in success with probability p and a failure with probability 1 - p.What is the probability that at least one success occurs in the first n trials?

A hydraulic structure has four gates which operate independently. The probability of failure off each gate is 0.2. Given that gate 1 has failed, the probability that both gates 2 and 3 will fail is

For the events A and B to be independent, the probability that both occur is 1/6 and probability that neither of them occur is 1/3. Then the probability of occurrence of A is

A program consists of two modules executed sequentially. Let f1 (t) and f2 (t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams