So I start with five batteries. The flashlight is assumed to require two batteries to shine. Each battery lifetime in use are independent random variables that are $\sim Exp(\lambda = 1)$. (Reserve batteries do not deteriorate). The question asks me to find the expectation of the flashlight lifetime.
My attempt: I call $T$ to be the lifetime of flashlight. Denote $B_{i}$ for $i = 1,2,3,4,5$ to be the lifetime of battery in use each $B_{i} \sim Exp(\lambda = 1)$.
I was thinking: If I choose the first two batteries, 1st battery runs out then we will have the 3rd with the 2nd. 2nd runs out, there are 3rd and 4th. Then third runs out we have 4th and the 5th. If 4th runs out, then we cannot use the flashlight. So, it has to be amount of time the 4th battery runs.
But I do not know how to express these in probability!! I would appreciate any help.