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Chapter 5: Problem 16

Christmas lights are often designed with a series circuit. This means that when one light burns out the entire string of lights goes black. Suppose that the lights are designed so that the probability a bulb will last 2 years is \(0.995 .\) The success or failure of a bulb is independent of the success or failure of other bulbs. (a) What is the probability that in a string of 100 lights all 100 will last 2 years? (b) What is the probability that at least one bulb will burn out in 2 years?

Short Answer

Expert verified
The probability that all 100 bulbs will last 2 years is approximately 0.6058. The probability that at least one bulb will burn out in 2 years is approximately 0.3942.

Step by step solution

01

Identify the Given Probability

The probability that a single bulb lasts 2 years is given as \( P(\text{lasting 2 years}) = 0.995 \).
02

Determine the Probability for All 100 Bulbs

The probability that all 100 bulbs will last 2 years is the product of the probability that each individual bulb lasts 2 years. This can be calculated using the equation: \[ P(\text{all 100 bulbs lasting 2 years}) = (0.995)^{100} \]
03

Calculate the Exponential Probability

Now, compute \( (0.995)^{100} \). Using a calculator or a computational tool, we find \[ (0.995)^{100} \approx 0.6058 \].
04

Answer for Part (a)

So, the probability that all 100 bulbs will last 2 years is approximately 0.6058.
05

Determine the Probability of At Least One Bulb Burning Out

The probability that at least one bulb will burn out in 2 years is the complement of the probability that all 100 bulbs will last. This can be calculated using: \[ P(\text{at least one bulb burns out}) = 1 - P(\text{all 100 bulbs last 2 years}) \]
06

Calculate the Complement for Part (b)

Using the previously calculated value for \( P(\text{all 100 bulbs lasting 2 years}) \approx 0.6058 \), we find: \[ P(\text{at least one bulb burns out}) = 1 - 0.6058 = 0.3942 \]
07

Answer for Part (b)

So, the probability that at least one bulb will burn out in 2 years is approximately 0.3942.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a measure of the likelihood of an event happening. In this exercise, the probability that a single Christmas light bulb will last for 2 years is given as 0.995.
This means that each bulb has a 99.5% chance of lasting 2 years.
We can assign a probability value, between 0 and 1, to any event. A probability of 1 means the event is certain to happen, whereas a probability of 0 means it will definitely not happen.

To find the probability of all 100 bulbs lasting for 2 years, we use the rule for the probability of independent events.
Independent Events
Independent events are events that do not affect each other鈥檚 outcomes. In this exercise, the success or failure of one bulb does not influence the success or failure of another bulb.
This characteristic allows us to multiply their individual probabilities to find the overall probability of a combined event.
For instance, if the probability of one bulb lasting is 0.995, the probability of a second, independent bulb also lasting is another 0.995.
The overall probability for all 100 bulbs lasting 2 years is: One hundred of these occurrences would be: The calculation becomes: Using a calculator, we get: almost 60.58%.
Complement Rule
The complement rule in probability states that the probability of an event not occurring is 1 minus the probability that the event does occur.
In the case of the Christmas lights, to find the probability that at least one bulb will burn out in 2 years, we need to first calculate the probability that all 100 bulbs will last 2 years.
Once we have that, we subtract this probability from 1 to find the complement.
From our previous steps, we know the probability that all 100 bulbs will last is approximately 0.6058. Using the complement rule: At least one bulb burning out: 0.3942.

So, there is about a 39.42% chance that at least one bulb will fail in 2 years.

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