Problem 25 - Algebra and Trigonometry

Question

Defective Light Bulbs The DimBulb Lighting Company manufactures light bulbs for appliances such as ovens and refrigerators. Typically, 0.5\(\%\) of their bulbs are defective. From a crate with 100 bulbs, three are tested. Find the probability that the given event occurs. (a) All three bulbs are defective. (b) One or more bulbs is defective.

Step-by-Step Solution

Answer

(a) 0.000000125; (b) 0.014925125.

Step 1: Understanding the Problem

DimBulb Lighting Company produces bulbs with a defect rate of 0.5\(\%\). From a batch of 100 bulbs, we want to find specific probabilities based on outcomes of testing three bulbs. We will solve for two cases: (a) all three bulbs are defective, and (b) at least one bulb is defective.

Step 2: Finding the Probability of a Defective Bulb

The probability that a single bulb is defective is 0.5\(\%\), which is 0.005 in decimal form.

Step 3: Calculating the Probability for Part (a)

To find the probability that all three tested bulbs are defective, use the multiplication rule for independent events. Since the probability of one being defective is 0.005, the probability for all three is: \[(0.005)^3 = 0.000000125\]

Step 4: Calculating the Probability of a Non-Defective Bulb

If 0.5\(\%\) are defective, then 99.5\(\%\) are non-defective. In decimals, this probability is 0.995.

Step 5: Solving for Part (b)

To find the probability of one or more bulbs being defective, it's easier to calculate the probability of the complementary event, which is all three bulbs being non-defective: \[(0.995)^3 = 0.985074875\] Then subtract this from 1: \[1 - 0.985074875 = 0.014925125\]

Step 6: Final Calculations

Both probabilities are very small, indicating that seeing even one defective bulb is a rare event in small samples. The calculations for the correct probabilities are: (a) 0.000000125, and (b) 0.014925125.

Key Concepts

Defective Rate

When discussing defective rates in probability theory, we focus on understanding the likelihood that a product, such as a light bulb, doesn't meet quality standards. In our example, DimBulb Lighting Company has a known defective rate of 0.5\(\%\). This means out of every 100 bulbs, on average, 0.5 bulbs are expected to be faulty.

Converting the percentage to a decimal is vital for probability calculations. The defective rate becomes 0.005 in decimal form, representing the probability that any randomly selected bulb from the batch will be defective. This rate serves as a foundational probability in determining outcomes in other events. In practical terms, understanding the defective rate helps businesses in quality assurance and cost management.

A few points to note about defective rates:

• Provides insights into product quality and manufacturing consistency.
• Essential for calculating probabilities in quality assessments.
• Helps companies to improve processes and reduce waste.

Multiplication Rule for Independent Events

The multiplication rule for independent events is a powerful tool in probability theory. It allows us to find the combined probability of two or more independent events happening. Two events are independent if the occurrence of one does not affect the occurrence of the other.

In the light bulb scenario, testing each bulb is an independent event since the outcome of one test doesn't influence the others. To calculate the probability that all three tested bulbs are defective, we multiply the individual probabilities of each bulb being defective:

\[(0.005)^3 = 0.000000125\]

This small probability indicates that it’s a rare occurrence for all three bulbs to be defective. Understanding and using the multiplication rule is crucial in scenarios where multiple independent events are involved:

• Can calculate combined probabilities for multiple outcomes.
• Requires knowledge of each event's individual probability.
• Frequently used in quality control and risk assessment.

Complementary Probability

Complementary probability is a useful concept for deriving the probability of at least one of multiple possible outcomes occurring by first determining the probability of the opposite event. In simpler terms, to find the probability of at least one defective bulb, we first calculate the probability that none are defective.

For the non-defective bulbs, the probability is 0.995, or 99.5\(\%\). Using the multiplication rule for independent events, calculate the probability that all three bulbs tested are non-defective:

\[(0.995)^3 = 0.985074875\]

The complementary probability for at least one defective bulb is then:

\[1 - 0.985074875 = 0.014925125\]

Thus, there's approximately a 1.49\(\%\) chance that at least one bulb is defective. This method simplifies complex probability calculations and is especially helpful when dealing with multiple outcomes:

• Focuses on easier calculations for complementary events.
• Helpful in assessing risks and likelihood in varied scenarios.
• Often leads to more straightforward solutions in probability exercises.