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 0.98 of working for 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.
The Correct Answer and Explanation is:
To find the probability that the entire string of 20 Christmas lights stays lit for 3 years, we need to calculate the probability that all 20 lights work during that period.
Each light has a 0.98 probability of working for 3 years, and the lights are wired in series, meaning if any one fails, the entire string fails.
Because the lights fail independently, the probability that all 20 lights work is the product of their individual probabilities: P(All 20 lights work)=(0.98)20P(\text{All 20 lights work}) = (0.98)^{20} P=0.9820≈0.6676P = 0.98^{20} \approx 0.6676
✅ Final Answer: 0.6676 (or about 66.76%)
📘 Explanation (300+ words):
This problem revolves around compound probability and independent events. Each of the 20 lights has a 98% chance of operating without failure for a 3-year period. Because they are connected in series, if even one light fails, the entire string goes dark. This is a classic reliability issue in systems engineering.
When events are independent, the probability of all events happening is the product of their individual probabilities. Here, we’re interested in the event that all 20 lights function properly, so we calculate: P=(0.98)×(0.98)×⋯(20 times)=0.9820P = (0.98) \times (0.98) \times \cdots \text{(20 times)} = 0.98^{20}
This results in: P=0.9820≈0.6676P = 0.98^{20} \approx 0.6676
This means there is about a 66.76% chance that all 20 lights will remain functional over the 3-year period. Conversely, there’s about a 33.24% chance that at least one light will fail, making the whole string go dark.
This type of scenario highlights a weakness of series circuits in real-life applications. Even though each individual component has a high reliability (98% is quite good), the overall system reliability drops significantly as more components are added. That’s why modern light strings often use parallel wiring, where one failed bulb does not affect the others.
In summary, due to the compounding effect of multiple independent failure points in a series system, even small failure probabilities can drastically reduce overall system reliability over time. This is an essential concept in both electrical design and probability theory.