You participate in a chocolate lottery, where every participant gets a bag. Your Christmas elf has filled the bags like this: he puts a dark chocolate into the bag. Then he tosses a coin. If the coin lands heads, he adds a milk chocolate, and tosses the coin again, and repeats the process. The process stops at the first tails. Let \(A_{1}\) be that there are zero milk chocolates in the bag, \(A_{2}\) that there is one, etc. Now, your elf is coming to you. You know how he filled the bag, but the bag is enchanted (of course), so the size does not reveal what's inside.

(a) Show that the prior probabilities are: \(P\left(A_{1}ight)=1 / 2\), whereas \(P\left(A_{2}ight)=\) \(1 / 4, P\left(A_{3}ight)=1 / 8, \ldots, P\left(A_{k}ight)=1 / 2^{k}\), which means that the prior probability distribution function is \(f(k)=1 / 2^{k}\) for all positive integers \(k\).

(b) The elf allows you to shout "Abra cadabra, chocolate come to me!" to your bag, and then a random chocolate from the bag will appear. When you try it, you get a dark chocolate. This is your observation \(B\). Show that the probability of getting a dark chocolate from bag \(k\) is \(P\left(B \mid A_{k}ight)=\) positive/total \(=1 / k\). This is your likelihood.

(c) Find the updated ( posterior) probability of the alternatives \(A_{k}\). (Hint: \(\left.\sum_{j=1}^{\infty} 2^{-j} \times j^{-1}=\ln (2).ight)\)