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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?
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A communication system consists of n components components. Each of th...
Five component system functions correctly if at least 3 components function and 3 component system component function correctly if at least two components function.
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A communication system consists of n components components. Each of th...
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
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
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A communication system consists of n components components. Each of th...
Five component system functions correctly if at least 3 components function and 3 component system component function correctly if at least two components function.
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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?
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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? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 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? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 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? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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