Question

a string of christmas lights contains 20 lights. the lights are wired so 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. what is the probability that the string of lights will not fail for the 3-year period?

Answer 1

The probability that a string of Christmas lights will not fail over a 3-year period is found by raising the probability that one bulb will not fail (0.98) to the power of the number of bulbs (20), resulting in a probability of approximately 0.667 or 66.7%.

Step-by-step explanation:

The question is about calculating the probability that a string of Christmas lights wired in series will last for a specific period without any of the lights failing. Since each light bulb has an independent probability of failing, and the whole string fails if any single light fails, we can approach this problem by considering the probability that none of the lights will fail over the specified time.

Given that each light bulb has a failure probability of 0.02 during a 3-year period, the probability that a single bulb will not fail is 1 - 0.02 = 0.98. Since there are 20 lights in the string, and they fail independently, the probability that all of them will not fail is the product of the individual probabilities for each bulb. Thus, it is 0.9820.

The calculation is as follows:

1. Calculate the probability that one bulb will not fail: 0.98.
2. Raise this probability to the power of the number of bulbs: 0.9820.
3. The result is the overall probability that the string of lights will not fail over the 3-year period.

Let's calculate it:

0.9820 ≈ 0.667.

Therefore, the probability that the string of lights will not fail for the 3-year period is approximately 0.667 or 66.7%.