🎲 1.4 - Probability and Probability Models
Key Question: How can engineers make confident decisions when working with only partial data (a sample)?
Probability models help quantify the risk of making incorrect decisions when only a sample of data is available. These models are essential for quality control, reliability engineering, and process monitoring.
Example Scenario – Wafer Inspection:
A semiconductor plant has 25 wafers in a batch. Only 1 is defective. If we inspect 3 wafers at random, what’s the chance we catch the bad one?
- If we detect the defective wafer, we save the batch.
- If not, the entire lot may be shipped with a defect – costly mistake!
Probability Models Help Engineers:
- Estimate the likelihood of detecting faults in a process.
- Decide how large a sample is needed for quality control.
- Use statistical distributions (like the normal distribution) to monitor changes in continuous measurements (e.g., temperature, concentration).
Case: Process Control Using Acetone Concentration
Suppose a chemical process is designed to maintain acetone concentration at a target level. Engineers take hourly samples. They need to determine whether any shift in concentration is due to natural variation or a real process issue.
Solution: Use control limits based on a probability distribution (e.g., normal) to flag abnormal shifts.
New Example – Emergency Room Staffing:
In a hospital ER, managers must decide how many nurses to schedule each shift. If they overstaff, resources are wasted. If they understaff, patient wait times grow and care may suffer.
Solution: Analyze past arrival patterns (e.g., average number of patients per hour) and fit a probability model, such as a Poisson distribution. Then use that model to predict likely demand ranges and optimize staffing.
- Probability helps estimate the risk of too many or too few nurses.
- Decision-makers can justify staffing plans with statistical evidence.
- Reduces emotional or arbitrary staffing decisions.
Why Probability is Critical:
- It supports statistical inference — drawing conclusions from a sample to a population.
- It quantifies uncertainty: What is the chance we are wrong?
- It lets engineers create rules for process adjustment that balance false alarms with missed detections.
Bottom Line:
Probability lets engineers make data-based decisions even under uncertainty. In Chapters 2–4, you’ll learn to apply these tools using distributions and sampling theory to control quality, reduce risk, and design better systems.