Memoryless Lights

Probability Level 3

James is having some issues wiring his house. He notices that regardless of the amount of time a lightbulb has been on, the probability of it dying out in the next twelve hours is exactly 0.2. James also hates doing chores at night (from 6 PM - 6 AM). If James puts in a lightbulb at 12 PM, what is the probability that James will have to replace the lightbulb between 6 PM - 6 AM on any day (disregard daylight-savings time)? Round your answer to 4 decimal places.

The answer is 0.4969.

1 solution

Patrick Corn
May 31, 2018

Let $p$ be the probability that a lightbulb stays on in a given six-hour span. Then $p^2 = 0.8.$

The probability that the lightbulb burns out between 6 PM and 6 AM is $\begin{aligned} p(1-p)+p^2(1-p)+p^5(1-p)+p^6(1-p)+p^9(1-p)+p^{10}(1-p) + \cdots &= p - p^3 + p^5 - p^7 + p^9 - \cdots \\ &= \frac{p}{1+p^2} = \frac{\sqrt{0.8}}{1.8} \approx \fbox{0.4969}. \end{aligned}$

Memoryless Lights

The answer is 0.4969.

1 solution

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