Problem 20 - Algebra 2 and Trigonometry

Question

The quality control department of a company that produces flashbulbs finds that 1 out of \(1,000\) bulbs tested fails to function properly. The flashbulbs are sold in packages of four. What is the probability that all the bulbs in a package will function properly?

Step-by-Step Solution

Answer

The probability that all four bulbs function properly is approximately 0.996.

Step 1: Define Probability of One Bulb Functioning

The probability that a single bulb functions properly is calculated by subtracting the probability of failure from one. Since 1 out of 1,000 bulbs fails, the probability of one bulb not failing is \(1 - \frac{1}{1000} = \frac{999}{1000}\).

Step 2: Calculate Probability of All Bulbs Functioning in a Package

A package contains four bulbs, and we need the probability that all four function properly. Since the functioning of each bulb is independent, multiply the probability of one bulb working properly four times: \((\frac{999}{1000})^4\).

Step 3: Perform the Final Calculation

Compute \((\frac{999}{1000})^4\) using a calculator. This results in approximately 0.996004. Therefore, the probability that all bulbs in a package will function properly is about \(0.996\).

Key Concepts

Understanding Independent Events

When we talk about independent events in probability, we refer to situations where the outcome of one event does not affect the outcome of another. In our flashbulb example, the performance of each bulb in a package is considered independent. This means that whether the first bulb works or not does not influence the likelihood that the second bulb will work—and so on for each subsequent bulb in the package.
This concept simplifies our calculations because we can multiply individual probabilities together without any additional conditions or considerations. Therefore, understanding that each flashbulb operates independently is crucial to calculate the probability correctly for them all functioning together.
When dealing with independent events, remember:

• Each event has no effect on the others.
• Probabilities can be multiplied to find the combined probability of all events occurring together.

Probability Calculation Basics

Calculating probability involves determining how likely an event is to occur, based on the information available. For any given event, the probability is determined by the ratio of favorable outcomes to the total number of possible outcomes.
In mathematical terms, if an event can happen in \[n\] ways out of a total of \[N\] possibilities, the probability \(P\) is expressed as:\[P = \frac{n}{N}\]For the flashbulbs, the probability calculation begins with knowing the failure rate. Since 1 out of 1,000 bulbs fails, the probability of a bulb functioning is:\[1 - \frac{1}{1000} = \frac{999}{1000}\]This calculation sets the stage for finding the collective probability of all bulbs in a package functioning properly, which involves the concept of independent events.
Always verify:

• Step-by-step breakdown of the scenario.
• Clear understanding of how each component of the event contributes to the overall probability.

Role of Quality Control in Probability

Quality control is essential in manufacturing to ensure consistent product performance. In our example with flashbulbs, quality control helps identify and reduce defects by keeping track of the failure rate. Knowing that 1 in 1,000 bulbs fails allows the company to predict and improve reliability over time.
Here’s how quality control and probability work hand-in-hand:

• Measurements from quality control (like failure rate) inform probability calculations.
• Accurate data from testing can help refine these probabilities and improve manufacturing processes.

Through rigorous quality checks, the company aims to better understand the likelihood of product defects, thus ensuring customer satisfaction and reducing costs associated with defective products. It illustrates how managing and improving quality directly impacts the overall probability of product success.

Probability of Success in Real-World Scenarios

The concept of probability of success refers to the likelihood that a series of events results in a favorable outcome. In the case of the flashbulbs, success means that all bulbs in a package work correctly.
To calculate this, we apply the probability of each bulb working (0.999) to all four bulbs in the package:\[\left(\frac{999}{1000}\right)^4 = 0.996004\]This results in a probability of approximately 0.996, or 99.6%, that a package will have all functioning bulbs.
In real-world scenarios, understanding the probability of success helps businesses set realistic expectations for product reliability. It can influence decisions on:

• Pricing models based on expected failure rates.
• Warranty terms and risk assessment.
• Continuous improvement efforts to enhance quality and success rates.

This practical application of probability enables better strategic decisions, operational efficiencies, and higher customer satisfaction.