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Chapter 5: Problem 89

Bright lights? A string of Christmas lights contains 20 lights. The lights are wired in series, so that if any light fails, the whole string will go dark. Each light has probability 0.02 of failing during a 3 -year period. The lights fail independently of each other. Find the probability that the string of lights will remain bright for 3 years.

Short Answer

Expert verified
The probability is approximately 0.6676.

Step by step solution

01

Identify the Probability of Success for One Light

Each light has a 0.02 probability of failing. Therefore, the probability of a light not failing (or remaining bright) for 3 years is the complement: \( P( ext{not failing}) = 1 - 0.02 = 0.98 \).
02

Determine Total Probability of Success for All Lights

Since the lights are wired in series, all 20 lights need to remain bright for the string to stay lit. The probability that all lights will remain bright is the product of the individual probabilities (assuming independence): \( (0.98)^{20} \).
03

Calculate the Probability

Compute the probability using the formula: \( (0.98)^{20} \), which results in approximately 0.6676. This is the probability that the string of Christmas lights will remain bright for 3 years.

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Key Concepts

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

Understanding Independent Events
In probability theory, independent events are events where the outcome of one event does not influence or change the outcome of another event. This is crucial when calculating the probability of multiple events happening sequentially. For instance, if you roll a die two times, the result of the first roll doesn't affect the second roll. They are independent events.
For the Christmas lights problem, each light bulb is considered an independent event. That means whether one light fails or not doesn't alter the probability of another light failing. This independence allows us to calculate the probability of all the lights remaining functional by simply multiplying the probability of each individual light staying bright over the specified duration.
Concept of Series Circuit in Bulbs
A series circuit connects multiple electrical components along a single path. In this setup, if there's a break at any point in the circuit (like a burned-out bulb), the entire circuit stops working, which is different from a parallel circuit where other paths might continue working.
In the given exercise, all 20 Christmas lights are connected in a series circuit. If any one of these bulbs were to fail, it would cause the entire string to go out. That's why it's important to ensure each bulb remains lit for the whole string to stay bright. The design exemplifies a series circuit dependency, where the failure of just one component affects the entire system.
Applying the Complement Rule
The complement rule in probability helps to find the probability of an event not occurring by subtracting the probability of the event occurring from 1. So, if the probability of a bulb failing is 0.02, the probability of the bulb not failing (or remaining operational) is the complement, calculated as:
  • \( P(\text{not failing}) = 1 - P(\text{failing}) = 1 - 0.02 = 0.98 \)
This formula is key in determining the probability of success for independent events. Once we know the chance of one light bulb operating well, we can then calculate the overall probability of all 20 bulbs operating, which in this case is
  • \((0.98)^{20}\)
By understanding the complement rule, you ascertain the safety margin of each component and ensure the entire system has a proactive support calculation for remaining functional over the given timeframe.

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