Agustus and Bill are working at a pledge drive. During a quiet period,
Agustus says 'Hey bill I'll bet you 2:1 that the phone rings in the next
minute.' (I.e. if it rings in the next minute, Bill pays Agustus $2, if not
Agustus pays Bill $1.) If the waiting time for the phone to ring next obeys
the exponential distribution:

f(x) {1/2e^-x/2 x>0
{ 0 otherwise

What is Agustus' expected profit (or loss) from this bet?

Apparently all I need integration for is to find the probability that the phone rings in the next minute. Then you have the probability that A wins, and you subtract from 1 to get
the probability that B wins. However I don't know what the limits are and how it works out. Please help.

Here is a hint:

For an exponential distribution,

P(t\leq T)=\int_{0}^{T}{\lambda}e^{{-\lambda}t}dt=1-e^{{-\lambda}T}

In this case, it looks as if T=1 and {\lambda}=\frac{1}{2}