Christmas Lights
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?

Verified step by step guidance

1. Identify the probability that a single bulb lasts 2 years, which is given as \$0.995$.
2. Recognize that the bulbs are independent, so the probability that all bulbs last 2 years is the product of the individual probabilities.
3. Since there are 100 bulbs, express the probability that all 100 bulbs last 2 years as \$0.995^{100}$.
4. Write the formula for the probability of all bulbs lasting 2 years as: \(P(\text{all 100 bulbs last}) = (0.995)^{100}\).
5. To find the final probability, you would calculate the value of \$0.995^{100}$ using a calculator or software.

Key Concepts

Independent Events

Independent events are those whose outcomes do not affect each other. In this problem, the lifespan of each bulb is independent, meaning the probability that one bulb lasts 2 years does not influence another bulb's lifespan. This allows us to multiply individual probabilities to find the combined probability.

Probability of Multiple Events Occurring

When multiple independent events all need to occur, the overall probability is the product of their individual probabilities. For 100 bulbs, each with a 0.995 chance of lasting 2 years, the probability all last is (0.995)^100, reflecting the combined likelihood of all successes.

Series Circuit Reliability

In a series circuit, the entire system fails if any single component fails. Here, if one bulb burns out, the whole string goes dark. This means the system's reliability depends on all bulbs functioning, emphasizing the importance of calculating the probability that every bulb lasts the specified time.