Problem 23 - 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, 3 are tested. Find the probability that the given event occurs. (a) All 3 bulbs are defective. (b) One or more bulbs is defective.

Step-by-Step Solution

Answer

(a) 0.000000125; (b) 0.014925125.

Step 1: Determine the Probability of a Defective Bulb

The probability of a bulb being defective is given as 0.5%. Convert this percentage to a decimal by dividing by 100. Therefore, the probability \( P(D) \) of a bulb being defective is \( 0.005 \), and the probability \( P(ND) \) of a bulb not being defective is \( 1 - 0.005 = 0.995 \).

Step 2: Calculate Probability for All 3 Bulbs Being Defective

The probability that all 3 bulbs are defective is found by multiplying the probability of one bulb being defective three times, since the events are independent: \( P(3D) = (0.005)^3 = 0.000000125 \).

Step 3: Calculate Probability for None of the 3 Bulbs Being Defective

To find the probability that none of the three bulbs are defective, use the probability of a bulb not being defective, \( 0.995 \), and multiply it three times: \( P(0D) = (0.995)^3 = 0.985074875 \).

Step 4: Calculate Probability for One or More Bulbs Being Defective

To find the probability that one or more bulbs are defective, use the complement of the probability that none are defective: \( P(\geq 1D) = 1 - P(0D) = 1 - 0.985074875 = 0.014925125 \).

Key Concepts

Defective Bulbs

Understanding the concept of defective bulbs is crucial when you're dealing with probability problems like the one involving light bulbs from DimBulb Lighting Company. A "defective bulb" refers to a bulb that fails to meet the desired operational standard. In simpler terms, it's a bulb that doesn't work as it should. In probability, knowing the chance of encountering a defective bulb is essential for making predictions about future outcomes.

In our context, 0.5% of the bulbs are defective. It's often easier to work with percentages when they're in decimal form. So, to determine how likely it is that a randomly chosen bulb from the batch is defective, convert the percentage to a decimal by dividing by 100, giving us a probability of 0.005. This tells us that if we pick a bulb at random from the crate, there is a 0.5% chance, or 0.005 probability, that it's defective.

Knowing this number helps in calculating more complex probabilities, such as whether all tested bulbs are defective or if there's at least one defective bulb in a subset of tested bulbs. Remember, every probability calculation depends heavily on understanding this initial figure.

Independent Events

In probability, the concept of independent events is key to understanding how likely multi-step or multi-event scenarios might occur. Two events are considered independent if the outcome of one does not impact the outcome of the other. Simply put, knowing what happens in one event doesn't change what we expect for the other.

Let's apply this to our light bulb problem. If you test one bulb and it turns out defective, it doesn't mean other bulbs are more likely to be defective. Testing each bulb is an independent event. This is crucial because it allows us to easily calculate multiple probabilities. For example, finding out the likelihood of all three tested bulbs being defective is a straightforward multiplication of the probabilities for each bulb.

If each bulb has a 0.005 probability of being defective, the chance that all three are defective is calculated by multiplying 0.005 by itself three times:

• First bulb: 0.005
• Second bulb: 0.005
• Third bulb: 0.005

This results in a probability of \[0.005 \times 0.005 \times 0.005 = 0.000000125\]. This multiplication only works because each bulb's condition is independent of the others.

Complement Rule

The complement rule is a fundamental principle in probability, which helps in calculating the probability of the occurrence of at least one event happening in a group of independent events. It revolves around the idea that the sum of probabilities of an event happening and it not happening equals 1.

In simpler terms, if you know the probability of something not happening, you can easily figure out the probability that it will happen at least once. In the case of the light bulb problem, to find the probability that at least one of the bulbs is defective, we first calculate the probability that none are defective, and then apply the complement rule.

This is done by using the probability of a bulb not being defective, which is 0.995. For three independent bulbs, multiply this probability by itself three times. This gives us \[(0.995)^3 = 0.985074875\]. This means the probability that none of the tested bulbs are defective is 0.985074875.

Using the complement rule, we can now find the probability that one or more bulbs are defective, i.e., \[1 - P( ext{none defective}) = 1 - 0.985074875 = 0.014925125\].So, there's approximately a 1.49% chance that one or more bulbs in the test will be defective. This method simplifies calculations and is very handy for probability problems involving multiple events.