A lamp has two bulbs, each of a type with an average lifetime of 4 hours. The probability density function for the lifetime of a bulb is .
What is the probability that both of the bulbs will fail within 4 hours?
i took the douple integral of this function with bounds from 0 to 4 and got the answer 1 which isn't right where did i go wrong?

2. Originally Posted by koalamath
A lamp has two bulbs, each of a type with an average lifetime of 4 hours. The probability density function for the lifetime of a bulb is .
What is the probability that both of the bulbs will fail within 4 hours?
i took the douple integral of this function with bounds from 0 to 4 and got the answer 1 which isn't right where did i go wrong?
Are the lifetimes of each bulb independent of each other? If so:

$\Pr(T < 4) = \int_{0}^{4} \frac{1}{4} e^{-t/4} \, dt = 1 - \frac{1}{e}$.

Therefore the probability of both bulbs failing within 4 hours is $\left(1 - \frac{1}{e}\right) \cdot \left(1 - \frac{1}{e}\right) = .....$

3. Thank you!