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.)

Verified step by step guidance

1. Step 1: Understand the arrangement of the surge protectors in series. When two components are arranged in series, both must work correctly for the television to be protected. This means the probability of protection is the product of the probabilities of each surge protector working correctly.
2. Step 2: Identify the probability of each surge protector working correctly. From the problem, each surge protector has a probability of 0.985 of working correctly during a voltage surge.
3. Step 3: Use the formula for the probability of both surge protectors working correctly in series: \( P_{series} = P_{p} \times P_{q} \), where \( P_{p} \) and \( P_{q} \) are the probabilities of surge protectors p and q working correctly.
4. Step 4: Substitute the given probabilities into the formula: \( P_{series} = 0.985 \times 0.985 \). This calculation will give the probability that a voltage surge will not damage the television.
5. Step 5: Interpret the result. The calculated probability represents the likelihood that both surge protectors will function correctly in series, ensuring the television is protected from a voltage surge.

Key Concepts

Probability of Independent Events

In probability theory, independent events are those whose outcomes do not affect each other. For example, if two surge protectors operate independently, the probability of both functioning correctly during a voltage surge can be calculated by multiplying their individual probabilities. This concept is crucial for determining the overall effectiveness of multiple protective devices in series.

Series Configuration

In a series configuration, devices are connected in such a way that the failure of one device affects the entire system. For surge protectors, this means that if one fails to work, the voltage surge can damage the television. Understanding this configuration is essential for calculating the overall probability of protection when multiple devices are used.

Complementary Probability

Complementary probability refers to the likelihood of an event not occurring, which is calculated as 1 minus the probability of the event occurring. In the context of surge protectors, to find the probability that the television is not damaged, one must consider the probability that at least one surge protector fails and subtract it from 1. This concept is vital for accurately assessing the risk of damage.