🎯 A Simple Trick for Computing Hit Probabilities

Why an old generating-function idea still beats brute force

You take n independent shots, and each shot has the same probability of hitting the target — call it p.
If you’ve ever taken an intro probability course, you’ve seen the result:

Easy, clean, elegant.
This is the binomial distribution.

But real life has a habit of ruining elegant formulas.

What if the shots don’t all have the same probability?

• The first shot is from 100 meters.

• The second from 80 meters.

• The third while holding your breath.

• The fourth after running.

Suddenly you have probabilities like:

p1=0.1,p2=0.2,p3=0.3,p4=0.4.

Now what’s the probability of getting 0 hits? 1 hit? 2 hits?

You can’t use the binomial formula anymore — the pi​’s are all different.
You could enumerate all combinations of hits and misses…
but with just 10 shots that’s already 2^10=1024 cases.

There has to be something better.

And there is.