I need some assistance in solving the following problem:
We are given a bag containing n unbiased coins. We are told that n − 1 of these coins are normal, that is, they have a head on one side and a tail in the other. The remaining one is fake and has heads on both sides. We are picking a coin from the bag uniformly at random.
We can devise the following method to determine if the coin is fake or not. We flip it k times, afer which we conclude that it is the fake one if all k flips have resulted in heads, else we conclude that it is normal. What is the probability that using this method we arrive at a wrong conclusion?
I declare the following events :
- F : the coin we picked is the fake one
- N : the coin we picked is normal (simply F complement)
- Hi : outcome of the i'th toss is a head
I though of simply solving this problem by calculating P[N | H1, ... , Hk] but it turns out we need to solve this problem using unconditional probability, and this is where I am stuck.