Problem 123 - Applied Statistics and Probability for Engineers

Question

The probability is \(1 \%\) that an electrical connector that is kept dry fails during the warranty pcriod of a portable computer. If the connector is ever wet, the probability of a failure during the warranty period is \(5 \% .\) If \(90 \%\) of the connectors are kept dry and \(10 \%\) are wet, what proportion of conncctors fail during the warranty period?

Step-by-Step Solution

Answer

1.4% of the connectors fail during the warranty period.

Step 1: Define the Probabilities

Let's define the probabilities given in the problem. Let \( P(F|D) \) be the probability of failure when a connector is kept dry, \( P(F|W) \) be the probability of failure when it is wet, \( P(D) \) be the probability that a connector is kept dry, and \( P(W) \) be the probability that it is wet. We are given: \( P(F|D) = 0.01 \), \( P(F|W) = 0.05 \), \( P(D) = 0.90 \), and \( P(W) = 0.10 \).

Step 2: Apply Total Probability Theorem

According to the Total Probability Theorem, the total probability of failure \( P(F) \) can be calculated as the weighted sum of the failures for connectors that are dry and wet. Thus, \[ P(F) = P(F|D) \cdot P(D) + P(F|W) \cdot P(W) \].

Step 3: Calculate Each Component

Calculate the contribution to failure for each type of connector: \( P(F|D) \cdot P(D) = 0.01 \times 0.90 = 0.009 \) and \( P(F|W) \cdot P(W) = 0.05 \times 0.10 = 0.005 \).

Step 4: Sum the Contributions

Now, add the contributions from each type of connector to find the overall probability of failure: \[ P(F) = 0.009 + 0.005 = 0.014 \].

Step 5: Interpret the Result

The calculated probability of \( 0.014 \) represents the proportion of connectors expected to fail during the warranty period out of the total number of connectors.

Key Concepts

Total Probability Theorem

The Total Probability Theorem is a fundamental concept in probability theory. It helps us find the probability of an event by considering all possible ways that event can occur. In simpler terms, it allows us to break down complex problems into smaller parts that are easier to handle. This way, we can calculate the overall probability by adding up the probabilities of all the individual scenarios.

Imagine you are trying to figure out the chance of an event happening, like a connector failing. The event can occur in multiple ways, such as if the connector is dry or wet. By splitting the problem into these scenarios, you can use known probabilities for each case to find the total probability.

In the given problem, since we know the probabilities of a connector failing when wet and when dry, we can apply the Total Probability Theorem to calculate the overall chance of failure. The formula we use is:

• Calculate the probability of failure if dry, multiplied by the probability of being dry.
• Calculate the probability of failure if wet, multiplied by the probability of being wet.
• Add them together to get the total probability of failure.

This makes it a very useful tool for dealing with situations involving different scenarios or states.

Conditional Probability

Conditional Probability focuses on finding the probability of an event given that another event has occurred. This concept is essential when the environment or situation impacts the probability of outcomes. When you hear 'conditional probability,' think of a dependent scenario.

In our exercise, conditional probability means understanding how the state of the connector, whether wet or dry, affects its failure rate. The probability of a failure is different for each condition—you wouldn't use the same probability for dry and wet connectors.

Here, you are given conditional probabilities:

• The probability ( P(F|D) ) a connector fails given it is kept dry, which is 0.01.
• The probability ( P(F|W) ) a connector fails given it is wet, which is 0.05.

You can see how these probabilities depend on the conditions, or what we also call the 'state' of the object involved.

Failure Analysis

Failure Analysis in the context of probability theory involves analyzing the scenarios that contribute to failures and calculating the overall chance of failure occurring. This process is crucial in many real-world applications, such as engineering and manufacturing, where knowing the likelihood of failure helps in designing more reliable systems.

In the exercise we solved, failure analysis was used to determine the overall probability that a connector will fail during the warranty period. This was done by examining each condition separately—when connectors are kept dry and when they are wet—and then combining these results according to the proportion of each condition occurring.

Through failure analysis, we don't just find which scenario has the highest failure probability but also consider how frequent or likely each scenario is. This means we can effectively predict the expected proportion of failures among a population of connectors by taking every relevant factor into account.