Particle Decay Homework: Prove P(t)=1 & Calculate Average Lifetime

[tomeatworld]

Homework Statement

We know that the probability of a particle decaying between time t=o and t=t is given by P(t)=e^-$$\Gamma$$t
a) Show the probability it decays in time between 0 and $$\infty$$ is 1
b) By considering decays at all possible times, calculate the average lifetime $$\tau$$ of the particle.

Homework Equations

As above

The Attempt at a Solution

Started just by saying that rate of decay = -$$\Gamma$$e^-$$\Gamma$$t but really nothing seems to be getting me to a decay of 1. Any hints would be muchly appreciated.
Edit: Ok, for the first part, I've argued it by saying that at a time t=0 the proability is 1 that the particle exists, while at a time t=$$\infty$$ the probability is 0 and so the change in probability is 1. Is that about right? Still can't get started with the second.

 

[NateD]

Edit2:Ok, I've got a similar argument for the second part, that the average rate of decay is \Gamma/2 so the average lifetime is \tau=2/\Gamma. Is this correct?