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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.

Philippe
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Then we search the probability to get a false conclusion, meaning to get $n$ consecutive heads ( 100% ) while it is a normal coin.

$P = \frac {1}{2^n}$

If you want to know to probability to get a wrong result in general, and if the ratio fake on all the coins is $f$ , you can count it like that

$P_g = \frac {1}{2^n} \times (1-f) + f \times 0 = \frac {1-f}{2^n}$