Problem 4 - Basic Engineering Mathematics

Question

The probability of a component failing in one year due to excessive temperature is \(\frac{1}{20}\), due to excessive vibration is \(\frac{1}{25}\) and due to excessive humidity is \(\frac{1}{50}\). Determine the probabilities that during a one-year period a component: (a) fails due to excessive temperature and excessive vibration, (b) fails due to excessive vibration or excessive humidity, and (c) will not fail because of both excessive temperature and excessive humidity.

Step-by-Step Solution

Answer

(a) \( \frac{1}{500} \), (b) \( \frac{13}{250} \), (c) \( \frac{999}{1000} \).

Step 1: Understanding the Events

Denote the events as follows: Let \( T \) be the event that the component fails due to excessive temperature with probability \( P(T) = \frac{1}{20} \). Let \( V \) be the event that the component fails due to excessive vibration with probability \( P(V) = \frac{1}{25} \). Let \( H \) be the event that the component fails due to excessive humidity with probability \( P(H) = \frac{1}{50} \).

Step 2a: Calculating Probability for Part (a)

For part (a), find the probability that the component fails due to both excessive temperature and excessive vibration. Assuming that these are independent events, the probability of both occurring is the product of their probabilities: \[ P(T \cap V) = P(T) \times P(V) = \frac{1}{20} \times \frac{1}{25} = \frac{1}{500} \].

Step 2b: Calculating Probability for Part (b)

For part (b), determine the probability that the component fails due to excessive vibration or excessive humidity. Use the formula for the union of two independent events: \[ P(V \cup H) = P(V) + P(H) - P(V \cap H) \]. First, calculate \( P(V \cap H) = \frac{1}{25} \times \frac{1}{50} = \frac{1}{1250} \). Then substitute to find the union: \[ P(V \cup H) = \frac{1}{25} + \frac{1}{50} - \frac{1}{1250} = \frac{5}{125} + \frac{2.5}{125} - \frac{1}{125} = \frac{6.5}{125} = \frac{13}{250} \].

Step 2c: Calculating Probability for Part (c)

For part (c), find the probability that the component will not fail due to both excessive temperature and excessive humidity. First, find the probability that it fails due to both: \[ P(T \cap H) = \frac{1}{20} \times \frac{1}{50} = \frac{1}{1000} \]. The probability of not failing due to both is: \[ 1 - P(T \cap H) = 1 - \frac{1}{1000} = \frac{999}{1000} \].

Step 5: Conclusion

The probability for (a) is \( \frac{1}{500} \), (b) is \( \frac{13}{250} \), and (c) is \( \frac{999}{1000} \).

Key Concepts

Independent Events

Independent events in probability theory are those events where the occurrence of one event does not influence or affect the occurrence of another. This means the probability of both events happening together is simply the product of their individual probabilities. For example, if we take events T (excessive temperature) and V (excessive vibration), we assume that whether or not a component fails due to excessive temperature doesn't change the likelihood that it will fail due to excessive vibration.
For independent events, if event A has a probability of \( P(A) \) and event B has a probability of \( P(B) \), then the probability of both events occurring (A and B) is given by:

• \( P(A \cap B) = P(A) \times P(B) \)

In our exercise, to compute the probability of component failure due to both excessive temperature and vibration, we utilize this formula, considering each event's probability individually. This simplification is a crucial concept when dealing with scenarios of multiple independent conditions.

Probability Calculations

Understanding how to perform probability calculations is fundamental in analyzing real-world scenarios. All probability values lie between 0 and 1, where 0 indicates impossible events and 1 indicates certain events. In discrete probability, we often deal with finite sample spaces where each outcome is equally likely.
To calculate probabilities correctly, it's important to define the events clearly and ascertain the relationships between them. Consider the probabilities from our exercise:

• \( P(T) = \frac{1}{20} \)
• \( P(V) = \frac{1}{25} \)
• \( P(H) = \frac{1}{50} \)

These numerical values are utilized directly to calculate complex probabilities like the intersection or union of events. Calculations often involve arithmetic operations such as addition, multiplication, or subtraction, dependent on the nature of the event relationship. The key is remembering that for independent events, probabilities multiply, while for the union, a combination of addition and subtraction (for overlap considerations) is used.

Union of Events

The union of events in probability refers to the probability of either one or more events happening at the same time. This can be visualized as the total area covered by the shadows of each event on a probability space.
The formula for the union of two events A and B is:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

This equation accounts for the intersection \( P(A \cap B) \) being subtracted once because it is counted in both \( P(A) \) and \( P(B) \). In our example, the component failure due to excessive vibration or humidity requires us to calculate:\( P(V \cup H) \), using the individual probabilities of V and H, and subtracting the overlap to avoid double-counting.
This method allows us to precisely calculate the likelihood that at least one of the specified conditions will cause failure—a key aspect in risk assessments and planning within engineering and technology fields.