When probability becomes non-sensical

You enter your plush office this morning and are greeted with a phone call from your broker. He wants to play a game, and feels you are the right player for this little experiment. He explains the rules - you shall toss a coin and call for a heads or tail. If your call is correct, then i'll double the money in your wallet. However, if you call wrong then i'll halve the money. If you dont participate in this game, the eccentric broker would give you 100 rupees.

What he doesn't know is that he is dealing with a mathematics stud from your school days and such problems are small-fry for you. Being the champion of probability, you pick your pen and try to simulate the problem ...
a) You assume that you hold 1000 rupees in your wallet
b) If you call right, the 1000 rupees become 2000 rupees.
c) If you call wrong, the 1000 rupees is down to 500 rupees.
d) In both cases, the probability of the event happening is 0.5

Simple equation .... X = [probability of (1000)*two + probability of (1000)*half] - 1000
or, X = [(0.5)*2000 + (0.5)*500] - 1000
or, X = [1000+ 250] - 1000
or, X = 250

Brilliant, you say. X is 250 rupees which is greater than 100 rupees (for non-participation). So, I must participate. Its a clear arbitrage of 150 rupees.

Think a little further .... You've participate in the game based on this probability analysis. What next? .... The use of probability analysis to aid you in decision making of participation in the game is only secondary. What's more important is how probability helps you in arriving at the correct call so that you can double your money.

To illustrate, imagine you had called wrong. Your purse would have been lighter by 500 rupees, although your profound mathematical skills would beg to differ on your interpretations of loss.

OK, think still further ... the broker has tossed the coin which lands on his palm. He hasn't shown you the result. Remember, you have used your probability skills to arrive at the logic that you have 150 rupees of "free money" to be made by participating in the game. He now gives you a second chance ... he says "I give you an option for you to not take this fall of the coin. You can tell me to toss the coin again.". You do your round of mathematics again and arrive at the same 150 rupees advantage. This still holds true. But what is the use?

By now you would have realised that the deal was inconsequential when we used probability analysis to the problem. Perhaps, just taking the 100 rupees was a better deal.

Adapted from "The Two Envelopes Paradox" by Keith Devlin