You have found a magic lamp on the beach. The genie in the lamp is a mathematician, and she has decided to fill a bag with rocks in the following way: she tossed a fair coin repeatedly until she got tails. She then counted the total number of tosses, \(X\), and filled the bag with \(2^{X}\) rocks. The first rock is an ordinary, gray pebble, but the rest are nuggets of gold.

(a) Let \(f(n)=P(X=n)\). Find an expression for the function \(f(n)\).

(b) Let \(g(n)\) be the probability that a random sampling from the bag yields a gold nugget, given that the genie tossed the coin \(n\) times. Find an expression for \(g(n)\).

(c) You now sample a rock from the bag, at random. It's a gold nugget. Find the updated probability distribution that the genie had flipped the coin \(n\) times.

(d) The genie puts a gray pebble in the bag as replacement for the gold nugget, and asks you to have a go one more time. What is the probability that the next rock is a gold nugget as well?