Question

Surge Protectors Refer to the accompanying figure showing surge protectors p and q used to protect an expensive television. If there is a surge in the voltage, the surge protector reduces it to a safe level. Assume that each surge protector has a 0.985 probability of working correctly when a voltage surge occurs.

a. If the two surge protectors are arranged in series, what is the probability that a voltage surge will not damage the television? (Do not round the answer.)

b. If the two surge protectors are arranged in parallel, what is the probability that a voltage surge will not damage the television? (Do not round the answer.)

c. Which arrangement should be used for better protection?

Short answer

a. The probability that a voltage surge will not damage the television if two surge protectors are connected in series is equal to 0.999775.

b. The probability that a voltage surge will not damage the television if two surge protectors are connected in parallel is equal to 0.970225.

c. As the surge protectors connected in series have a higher probability of not damaging the television; the series arrangement should be used for better protection.

Step-by-step solution

Given information

Two surge protectors are connected to protect the television from damage due to voltage surges.

The probability that a surge protector will work is 0.985.

Multiplication rule of probability 

For computingthe probability of two events that occur simultaneously, the individual probabilities of the two events are multiplied, considering the interdependency of the events being studied.

Compute the probability that a surge protector does not work

Let A and B be the events that the first and second surge protector work correctly, respectively.

The probability that a surge protector will work correctly is given to be equal to 0.985.

Here, the probability that a surge protector will not work correctly is given by:

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =\left(1-0.985\right) \\ =0.015 \end{gathered}$$

$$\begin{gathered} P\left(\overline{B}\right)=1-P\left(B\right) \\ =\left(1-0.985\right) \\ =0.015 \end{gathered}$$

Compute the probability that a surge protector works in the series configuration

a.

In a series arrangement, the television would not get damaged by a voltage surge if at least one of the two surge protectors works.

Three cases are possible:

• Both surge protectors are working.

$$\begin{gathered} P\left(AandB\right)=0.985\times 0.985 \\ =0.970225 \end{gathered}$$

• The first surge protector is working, but the second is not.

$$\begin{gathered} P\left(Aand\overline{B}\right)=0.985\times 0.015 \\ =0.14775 \end{gathered}$$

• The first surge protector is not working, but the second is working.

$$\begin{gathered} P\left(\overline{A}andB\right)=0.015\times 0.985 \\ =0.14775 \end{gathered}$$

The probability that the television will not get damaged due to a voltage surge if both the surge protectors are connected in series is the sum of the above three probabilities, and is given by:

$$\begin{gathered} P\left(nodamage\right)=0.970225+0.14775+0.14775 \\ =0.999775 \end{gathered}$$

Therefore, the probability that the television will not get damaged due to a voltage surge if both the surge protectors are connected in series is equal to 0.999775.

Compute the probability that a surge protector works in the parallel configuration

b.

In a parallel arrangement, the television would not get damaged by a voltage surge if both the surge protectors work.

The probability that both the surge protectors will work is given by:

$$\begin{gathered} P\left(AandB\right)=0.985\times 0.985 \\ =0.970225 \end{gathered}$$

Therefore, the probability that the television will not get damaged due to a voltage surge if both the surge protectors are connected in parallel is equal to 0.970225.

Analyze which arrangement protects better

c.

The arrangement which provides a greater probability of protecting the TV from voltage surge is regarded better.

As the probability of no damage to the television is high in the case of series arrangement (0.999775) as compared to the parallel arrangement (0.970225), the series arrangement is a better option for the protection of the television from a voltage surge