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A system consisting of n components functions if, and only if, at least one of n components functions. Suppose that all the n components of the system function independently, each with probability 3/4. If the probability of functioning of the system is 63/64, then the value of n is
- a)2
- b)4
- c)3
- d)5
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A system consisting of n components functions if, and only if, at leas...
Solution:
To find the value of n, we need to determine the number of components in the system.
Given:
- Each component functions independently with a probability of 3/4.
- The system functions if and only if at least one component functions.
- The probability of the system functioning is 63/64.
Let's break down the problem step by step:
Step 1: Probability of the system not functioning
The probability of a single component not functioning is 1 - (3/4) = 1/4. Since all components function independently, the probability of the entire system not functioning is (1/4)^n.
Step 2: Probability of the system functioning
The probability of the system functioning is given as 63/64.
Step 3: Finding the value of n
We can use the complement rule to find the probability of the system not functioning. According to the complement rule, the probability of an event occurring is 1 minus the probability of the event not occurring.
So, the probability of the system not functioning is 1 - (63/64) = 1/64.
Substituting this into Step 1:
(1/4)^n = 1/64
Taking the logarithm of both sides:
log[(1/4)^n] = log(1/64)
Using the logarithmic properties, we can bring down the exponent:
n * log(1/4) = log(1/64)
Simplifying the logarithms:
n * (-2) = (-6)
Dividing both sides by -2:
n = (-6) / (-2)
n = 3
Therefore, the value of n is 3.
To find the value of n, we need to determine the number of components in the system.
Given:
- Each component functions independently with a probability of 3/4.
- The system functions if and only if at least one component functions.
- The probability of the system functioning is 63/64.
Let's break down the problem step by step:
Step 1: Probability of the system not functioning
The probability of a single component not functioning is 1 - (3/4) = 1/4. Since all components function independently, the probability of the entire system not functioning is (1/4)^n.
Step 2: Probability of the system functioning
The probability of the system functioning is given as 63/64.
Step 3: Finding the value of n
We can use the complement rule to find the probability of the system not functioning. According to the complement rule, the probability of an event occurring is 1 minus the probability of the event not occurring.
So, the probability of the system not functioning is 1 - (63/64) = 1/64.
Substituting this into Step 1:
(1/4)^n = 1/64
Taking the logarithm of both sides:
log[(1/4)^n] = log(1/64)
Using the logarithmic properties, we can bring down the exponent:
n * log(1/4) = log(1/64)
Simplifying the logarithms:
n * (-2) = (-6)
Dividing both sides by -2:
n = (-6) / (-2)
n = 3
Therefore, the value of n is 3.
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A system consisting of n components functions if, and only if, at least one of n components functions. Suppose that all the n components of the system function independently, each withprobability 3/4.If the probability of functioning of the system is 63/64,then the value of n isa)2b)4c)3d)5Correct answer is option 'C'. Can you explain this answer?
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