Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A lamp has two bulbs, each of a type with an average lifetime of 5 hours. The probability density function for the lifetime of a bulb is $\sigma(t)$, What is the probability that both of the bulbs will fail within 2 hours?

If the probability of an event occurring is given by:

$$\int_{0}^\infty \sigma(t)dt$$

Then is it true that for two independent events, the probability is given by


share|cite|improve this question
16 minutes. $ $ – Did Jan 17 '13 at 7:38
up vote 1 down vote accepted

The probability that the first bulb fails within $2$ hours is $\int_0^2 \sigma(t)\,dt$. For the probability that both bulbs fail, assuming independence, just square the answer for one bulb. This gives $\left(\int_0^2 \sigma(t)\,dt\right)^2$, which is not the same as the answer proposed in the OP.

Without knowing more about $\sigma(t)$, we cannot produce an explicit numerical answer. The fact that the mean is $5$ is not enough to determine the probability a single bulb has lifetime $\le 2$.

Possibly (but unreasonably) you are expected to assume the lifetime has exponential distribution.

Added: It looks as if one is expected to assume exponential distribution. This is not a good model. The exponential distribution is an appropriate model for the lifetime of things that die but do not age, like atoms of a radioactive substance. Lightbulbs age!

But if we assume exponential distribution with mean $5$, then the density function is $\frac{1}{5}e^{-t/5}$ (for $t\ge 0$). Integration shows that the probability the lifetime is $\le x$ is $1-e^{-x/5}$. So the probability both our bulbs are dead by time $2$ is $\left(1-e^{-2/5}\right)^2$.

share|cite|improve this answer
it does thank you very much – Cactus BAMF Jan 17 '13 at 7:18
@CactusBAMF: What do you mean by "it does"? Do you mean assume exponential? – André Nicolas Jan 17 '13 at 7:21
There is an exponential function associated with the problem. – Cactus BAMF Jan 17 '13 at 7:22

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.