For an exam preparation I'm trying to solve the following task:
Suppose we have two coins:
- one is fake ($100\%$ chance of tails) and one normal
We throw the coin $n$ times ?
Give the probability that the fake coin was used after we had $n$ (or to make it harder $k $ for $ \ k<n$ ) successes.
I looked into my lecture notes but could not find anything resembling this problem.
My question is:
Can I get some hints, tips and literature so I will be able to solve this task ?
Consider the normal coin. Use the binomial distribution to find the probability we get at least one $heads$ in the normal coin out of $n$ flips. That will be 1 minus the probability we get all $tails$ or $1 - (1/2)^n$. The chance that you get such a $heads$ is thus the probability you choose the normal coin ($1/2$) times the probability you get at least a single $heads$. Can you finish this up?