Problem 30 - Algebra and Trigonometry

Question

Reliability of a Machine A machine that is used in a manufacturing process has four separate components, each of which has a 0.01 probability of failing on any given day. If any component fails, the entire machine breaks down. Find the probability that on a given day the indicated event occurs. (a) The machine breaks down. (b) The machine does not break down. (c) Only one component does not fail.

Step-by-Step Solution

Answer

(a) 0.0394, (b) 0.9606, (c) 0.000004.

Step 1: Identify the Probability of Component Failure

Each component of the machine has a probability of 0.01 of failing on a given day. This can be represented as \( P(F) = 0.01 \).

Step 2: Calculate the Probability of Component Success

The probability that a component does not fail (succeeds) is \( P(S) = 1 - P(F) = 1 - 0.01 = 0.99 \).

Step 3: Calculate the Probability of Machine Breakdown (Part a)

For the machine to break down, at least one of the components must fail. To find the probability that the machine breaks down, we first find the probability that none of the components fail. The probability that all four components do not fail is \( (0.99)^4 \). Therefore, the probability of at least one failing is \[ P(\text{Breakdown}) = 1 - (0.99)^4. \]

Step 4: Simplify Breakdown Probability Calculation

Calculate \((0.99)^4 = 0.96059601\), thus \[ P(\text{Breakdown}) = 1 - 0.96059601 = 0.03940399. \]

Step 5: Calculate Probability that Machine Does Not Break Down (Part b)

The probability that the machine does not break down is the probability that all components do not fail, which is \[ P(\text{Not Breakdown}) = (0.99)^4 = 0.96059601. \]

Step 6: Calculate Probability That Only One Component Succeeds (Part c)

For only one component to succeed, three must fail. The probability for this scenario can be calculated using the combination formula: \[ P(\text{Only One Succeeds}) = \binom{4}{3} \times (0.01)^3 \times 0.99. \]Calculating further, we get:\[ \binom{4}{3} = 4, \]\[ 4 \times (0.01)^3 \times 0.99 = 4 \times 0.000001 \times 0.99 = 0.00000396 \approx 0.000004. \]

Key Concepts

Understanding Component Reliability

Component reliability is fundamentally about ensuring that each part of a system continues to function as expected over time. In the context of this exercise, we are referring to the machine components, each with a 0.01 probability of failure on any given day. To calculate reliability, we need to assess both failure and success probabilities for these components.
Reliability is vital because even a small chance of failure can lead not just to a single part but the entire machine failing. We calculate the reliability of each component by determining the probability it does not fail, expressed as one minus the failure probability. Here, this probability, also known as component success, is 0.99 or 99%.
By focusing on component reliability, engineers can anticipate potential breakdowns and implement maintenance strategies to minimize downtime.

Mechanics of Machine Breakdown

Machine breakdown occurs when one or more components fail. The probability of a machine breakdown is a cumulative process of calculating the likelihood that at least one component ceases to function properly.
To find this probability, it is often easier to first calculate the probability that none of the components fail, which can be represented by the power of probability of success of each component over the total number of components. In this case, the probability of all components not failing is calculated as \[ (0.99)^4 \] which equals approximately 0.9606. Thus, the probability of at least one component failing—triggering a machine breakdown—is \[ 1 - (0.99)^4 = 0.0394. \]
Understanding this process is crucial; it illustrates how dependent systems can be on each part functioning correctly. It also stresses the importance of robust components and adequately planned checks.

Probability Calculations Simplified

Probability calculations involve assessing the likelihood of various outcomes. In a scenario like this machine breakdown, various calculations help predict different outcomes:

• The breakdown probability: Getting this involves first calculating no failure probability and subtracting it from one.
• No-breakdown probability: This is simply the probability of each component not failing multiplied together, as shown with the calculation \[ (0.99)^4 = 0.9606. \]
• Only one component success: Here, we use combinations to calculate the scenario where one specific event occurs, with others failing. For one component to succeed, three must fail. This is a specific kind of probability calculation, utilizing the combination formula: \[ \binom{4}{3} \times (0.01)^3 \times 0.99, \] which gives a very slim chance of approximately 0.000004.

Mastering these calculations helps you predict and control the various possible operational states of the machinery, ensuring better management and maintenance.