Question

A smoke-detector system uses two devices, \(A\) and \(B\). If smoke is present, the probability that it will be detected by device \(A\) is \(.95 ;\) by device \(B, .98 ;\) and by both devices, \(.94 .\) a. If smoke is present, find the probability that the smoke will be detected by device \(A\) or device \(B\) or both devices. b. Find the probability that the smoke will not be detected.

Short answer

Answer: The probability of detecting smoke using either device A or device B or both is 0.99, and the probability of not detecting smoke at all is 0.01.

Step-by-step solution

Understand the given probabilities

We're given the following probabilities: - The probability that device A detects smoke, \(P(A) = 0.95\) - The probability that device B detects smoke, \(P(B) = 0.98\) - The probability that both devices detect smoke, \(P(A \cap B) = 0.94\)

Apply the addition rule for probability

To find the probability that smoke will be detected by device A or device B or both devices, we use the addition rule for probability, which states that \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Using the given probabilities, we can calculate this: \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.95 + 0.98 - 0.94 = 0.99\)

Find the probability of not detecting smoke

To find the probability that the smoke will not be detected, we need to find the complement of \(P(A \cup B)\). The complement of an event is 1 minus the probability of that event occuring: \(P(\text{Not detected}) = 1 - P(A \cup B) = 1 - 0.99 = 0.01\) So the answers are as follows: a. The probability that the smoke will be detected by device A or device B or both devices is \(0.99\). b. The probability that the smoke will not be detected is \(0.01\).

Key concepts

Addition Rule for Probability

Understanding the addition rule for probability is essential when dealing with multiple events and their chances of occurring. It is particularly useful when you want to know the likelihood that at least one of several different events will happen.

In our smoke-detector scenario, we have two independent devices, A and B, each with its own probability of detecting smoke. What if we need to know the probability that either device A or B or both will detect the smoke? Here, the addition rule comes in handy. The rule states that the probability of either event A or event B occurring (or both) can be calculated by adding the probabilities of each event individually and then subtracting the probability that both events occur simultaneously.

The general form of the addition rule is \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), where \(P(A \cup B)\) represents the probability of either event A or B occurring, \P(A)\ and \P(B)\ are the probabilities of events A and B independently, and \(P(A \cap B)\) is the probability of both events occurring together. This adjustment is necessary to avoid double-counting the overlap that both A and B detect.

Probability of Detection

The probability of detection measures how likely it is that a system or device will successfully identify a condition or event, such as smoke in a detector system's case. In the given exercise, two devices are capable of detecting smoke, and each has its own detection probability.

The probability of detection is critical in safety systems like smoke alarms because it helps to assess the reliability and efficiency of the system. With higher probabilities of detection, as we have with devices A and B (0.95 and 0.98, respectively), we trust the system more. Yet, knowing the probability of both devices detecting smoke, which is 0.94, implies a very high level of redundancy and reliability in our system. The higher the redundancy, the lower the likelihood that a failure to detect will occur, which is a crucial aspect of system safety design.

Complementary Probability

Complementary probability is an intuitive yet powerful concept in probability theory, providing a way to determine the likelihood of an event not happening based on the known probability of the event occurring. Simply put, it's the 'other side of the coin' when considering probabilities.

In practical terms, if the probability that smoke detectors will detect smoke is 0.99, the complementary probability represents the chance that these detectors will fail to detect it. Calculating the complement is straightforward: subtract the event's probability from 1. So, in our example, \(P(\text{Not detected}) = 1 - P(A \cup B) = 1 - 0.99 = 0.01\). Knowing the complementary probability is just as important as the probability of detection itself, especially when evaluating the potential risks and the necessary precautions for ensuring safety.