Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Before being allowed to enter a maximum-security area at a military installation, a person must pass three independent identification tests: a voice-pattern test, a fingerprint test, and a handwriting test. If the reliability of the first test is \(97 \%\), the reliability of the second test is \(98.5 \%\), and that of the third is \(98.5 \%\), what is the probability that this security system will allow an improperly identified person to enter the maximumsecurity area?

Step-by-Step Solution

Answer

The probability that the security system will allow an improperly identified person to enter the maximum-security area is approximately 6.23%. This is calculated by finding the complementary probability of all three independent identification tests correctly identifying a person (0.97 × 0.985 × 0.985 ≈ 0.937694725), and then subtracting this value from 1 (1 - 0.937694725 ≈ 0.062305275).

Step 1: Calculate the probability of all tests correctly identifying a person

First, we will find the probability that all three tests correctly identify someone. Since the tests are independent, we can simply multiply the reliability (probability of correct identification) of each test: Probability Test 1 Passing = 97% = 0.97 Probability Test 2 Passing = 98.5% = 0.985 Probability Test 3 Passing = 98.5% = 0.985 Probability all tests pass = 0.97 × 0.985 × 0.985

Step 2: Calculate the complementary probability

Now we will find the complementary probability, which is the probability of at least one test failing: Complementary probability = 1 - Probability all tests pass

Step 3: Calculate the probability of at least one test failing

Using the results from step 1 and step 2, we can calculate the probability of at least one test failing, and therefore allowing an improperly identified person to enter the maximum-security area: Probability of at least one test failing = 1 - (0.97 × 0.985 × 0.985) = 1 - (0.937694725) ≈ 0.062305275 So, the probability that the security system will allow an improperly identified person to enter the maximum-security area is approximately 6.23%.

Key Concepts

Independent Events

In probability theory, independent events are those events where the occurrence of one event does not affect the likelihood of another event occurring. This means that these events are not influenced by each other's outcomes at all.
For example, in the provided exercise, each of the identification tests – the voice-pattern test, the fingerprint test, and the handwriting test – are independent of one another. This indicates that the outcome of one test does not influence the outcome of the others.
Understanding the concept of independent events is crucial because it allows us to use a simple multiplication rule to calculate the probability of multiple independent events all occurring. Specifically, if you have several independent events, the probability that all of them occur simultaneously can be found by multiplying the probabilities of each event. In mathematical terms, if you have three independent events A, B, and C with probabilities \( P(A) \), \( P(B) \), and \( P(C) \), then the probability that all three events occur is \( P(A \cap B \cap C) = P(A) \times P(B) \times P(C) \).

Complementary Probability

Complementary probability deals with the likelihood of an event not occurring. It is essentially the converse of the original event's probability. If the probability of an event occurring is \( P(E) \), the complementary probability, or the probability of the event not occurring, is \( 1 - P(E) \).
In the exercise, we calculated the probability that all identification tests correctly identify a person. However, to solve the problem, we needed to find the probability of an incorrectly identified person gaining access, meaning at least one test fails.
Identifying the complementary probability involves calculating \( 1 - P(\text{all tests pass}) \). This technique is particularly useful in scenarios where it is easier to calculate the probability of an event happening rather than it failing directly.
Complementary probability is a powerful concept as it can simplify complex calculations, by finding the probability of the "opposite" scenario, and using it to answer the original question.

Mathematical Reliability

Reliability in a mathematical sense refers to the probability that a system or component performs its intended function under specific conditions for a designated period of time. It is an important measure in scenarios where maintaining consistent performance is critical.
In the given exercise, the reliability of each identification test represents the probability that it functions correctly in identifying the person.

• The voice-pattern test has a reliability of \(97\%\), meaning it correctly identifies the person 97 times out of 100 on average.
• The fingerprint and handwriting tests each have a reliability of \(98.5\%\).

Understanding reliability helps us gauge the overall effectiveness of a system composed of independent components, like the security system in the exercise. By determining the reliability of each component, we can predict the system’s chances of failure, which is the complementary probability discussed earlier, critical when designing systems where failure can have significant consequences.