# instantaneous probability and probability per unit time

Ok, I have a question and I have some vague thought of a correlation to probability density perhaps but I'm not sure how

Let's say there is some event that has a fixed probability of occurring at any given point in time. Let's call it a radioactive particle decaying. Then, at any given instant it could decay with a specific chance of it happening, correct?

Then I think it would be reasonable for the probability of it decaying over a specific time period would be the integral of the probability as a function of time divided by it's time period t. Then, since the function is constant, one would expect that it would just cancel back out to being exactly that constant probability of decaying at any point in time. So the probability of decaying in the next second is the same as decaying in the next year.

Is it then therefore possible at all to generate some time dependent probability function for decay for let's say an infinite amount of these particles which predicts how many particles decay in a given unit of time?

I know this follow up question isn't really math, but, if such a function cannot exist, then where does my model fail in describing real life radioactive decay?

Radioactive decay is modelled using the exponential distribution $$f(t)=\lambda e^{-\lambda t}$$. The quantity $$\frac{1}{\lambda}$$ is the half-live. The probability that the decay will happen in the first $$t_1$$ seconds is given by $$\int^{t_1}_0\lambda e^{-\lambda t}dt$$. If you let $$t_1$$ go to infinity then the probability of decay approaches one as it must.