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Problem 5 The five switches in the figure operate independently. Each switch i...

Sep 4, 2024

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Solution by Steps

step 1

The probability that the signal at the input will not be received at the output occurs when all switches are open. Since each switch is open with probability

(1-p)

, the probability that all five switches are open is given by:

(1-p)^5

step 2

Therefore, the probability that the signal is not received at the output is:

P(\text{not received}) = (1-p)^5

step 3

For the conditional probability that switch

S_4

is open given that the signal is received at the output, we first find the probability that the signal is received. The signal is received if at least one switch is closed. The probability that at least one switch is closed is:

1 - (1-p)^5

step 4

The conditional probability can be expressed as:

P(S_4 \text{ is open} | \text{signal received}) = \frac{P(S_4 \text{ is open and signal received})}{P(\text{signal received})}

The probability that

S_4

is open and the signal is received is the same as the probability that the other four switches are closed, which is

(1-p)^4 p

. Thus, we have:

P(S_4 \text{ is open} | \text{signal received}) = \frac{(1-p)^4 p}{1 - (1-p)^5}

Answer

The probability that the signal is not received is

(1-p)^5

and the conditional probability that switch

S_4

is open given that the signal is received is

\frac{(1-p)^4 p}{1 - (1-p)^5}

Key Concept

Understanding independent events and conditional probability in probability theory

Explanation

The solution involves calculating the probabilities of independent events and applying the definition of conditional probability to find the required probabilities.