# Terrible Probability

Kelvin is offered the following gamble: he is to choose a coin at random from a large collection of coins and toss it randomly. 8/21 of the coins in the collection are loaded towards a head (LH) and are 13/21 loaded towards a tail. If a coin is loaded towards a head, then when the coin is tossed randomly, there is a 3 /4 probability that a head will turn up and a 1/4 probability that a tail will turn up. If the coin is loaded towards tails, then there is a chance of tossing a tail 15/23 and there is a 8/23 probability that a head will turn up on any given toss. If Kelvin tosses a head, he loses $165, and if he tosses a tail, he wins $206. Kelvin is allowed to obtain "sample information" about the gamble. When he chooses the coin at random, he is allowed to toss it once before deciding to accept the gamble with that same coin. Suppose Kelvin tosses a head on the sample toss. Find Kelvin's expected gain/loss on the gamble if it is accepted. The solution is in negative abc*10^{-2}.