Problem 78 - Basic Statistics for Business and Economics with Student

Question

Two components, \(A\) and \(B\), are in series. (Being in series means that for the system to operate, both components \(A\) and \(B\) must work.) Assume the two components are independent. What is the probability the system works under these conditions? The probability \(A\) works is .90 and the probability \(B\) functions is also .90 .

Step-by-Step Solution

Answer

The probability that the system works is 0.81.

Step 1: Understand the Problem

We have two components, A and B, in series. For the overall system to work, both A and B must be operational. We need to calculate the combined probability that both components are functioning.

Step 2: Identify Given Probabilities

The probability that component A works is given as 0.90, and similarly, the probability that component B works is also 0.90.

Step 3: Calculate the Probability of Both A and B Working

Since components A and B are independent, the probability that both A and B work simultaneously is the product of their individual probabilities. Use the formula: \( P(A \text{ and } B) = P(A) \times P(B) \).

Step 4: Compute the Result

Substitute the given probabilities into the formula: \( P(A \text{ and } B) = 0.90 \times 0.90 \).

Step 5: Solve the Multiplication

Calculate the product: \( 0.90 \times 0.90 = 0.81 \). Therefore, the probability that both components work and the system operates is 0.81.

Key Concepts

Independent Events

Independent events are a crucial concept in probability and statistics, referring to situations where the occurrence of one event does not affect the occurrence of another. In this scenario, the two components, A and B, are independent.
This means that the performance of one component does not influence or change the probability of the other component's performance.

• For example, if component A works correctly, this provides no information about whether component B will work correctly and vice versa.
• Independence allows us to calculate the combined probability of both events occurring by multiplying their individual probabilities.

Understanding the independence of events is essential because it simplifies the calculation of probabilities in many real-world situations, such as reliability in systems, like the one described in the exercise.
Once you know events are independent, you can confidently apply multiplication to find combined probabilities.

Probability Calculation

Probability calculation can be likened to determining likelihood, a number between 0 and 1, where 0 means an event won't occur, and 1 means it definitely will.
To calculate the probability of independent events occurring together, we use multiplication. For the series system described:

• Given that the probability of component A working, \( P(A) \), is 0.90, and that of component B working, \( P(B) \), is also 0.90.
• The combined probability that both A and B work is given by multiplying these probabilities: \[ P(A \text{ and } B) = P(A) \times P(B) = 0.90 \times 0.90 \]
• This results in: \[ P(A \text{ and } B) = 0.81 \]

This calculation shows that despite each component having a high individual probability of functioning, the combined probability drops when both need to work simultaneously.

System Reliability

System reliability in a series system means ensuring the entire system functions as intended.
In systems like the one with components A and B, reliability is dependent on all parts working correctly. For series systems, if even one component fails, the whole system doesn’t work:

• The system reliability is directly linked to the probabilities of the individual components performing their function.
• Using the example provided, with each component having a reliability probability of 0.90, both needing to operate for the system to be reliable, gives an overall system reliability of 0.81.
• This indicates that while individual components may be reliable, series configurations mean multiplied reliability for the entire system.

Understanding this helps in designing systems, as engineers must decide on configurations that enhance reliability through redundancy or parallel components rather than series when possible.