# One shot events

Suppose you're renowned statistician down on his luck, for whatever reason, and only has \$100 to live on for the next week.

Suppose next, someone offers you a game by flipping a coin (no gimmicks with coin) and offers to give you 150 dollars if it lands on heads and take your 100 dollars if it lands on tails (no food for week).

Now you know no matter what the pay off the chances of no food remains the same (and being a renowned statistician you have no excuse not to know.)

How big does the expected pay off have to be before you'd bite and chance it on the flip of a coin? Can that decision be modelled with a statistical process or more a psychological one?

• (+1) no matter what, this is a great question since it can teach us a lot of how humans (are in general not be able to) deal with probabilities. – steffen May 2 '12 at 6:47
• may be connected to How best to communicate uncertainty? since some answers about how to deal with risks have been posted there – steffen May 2 '12 at 10:46

Check out prospect theory. In Daniel Kahneman's recent book Thinking, Fast and Slow, he discusses an example with coin flipping very much like this one, which served as a motivating example way back when prospect theory was being invented.

In this case, one is weighing the utility of extra money vs. the utility of being able to purchase food this week. I'd say the loss aversion towards going a week without food is substantial, and so empirically, the expected payoff would have to be very high indeed.

Some similar sorts of questions you might like:

(1) There are 500 people facing death from a building fire and as the building's fire marshal you have two options. In option (A) there is a 90% chance that you save all 500, but a 10% chance that everyone dies. In option (B) there is a 100% chance that exactly 400 people live and 100 people die. Which do you choose? What if a loved one is among the 500?

(2) Same as above, but your estimates were actually off. It turns out that in option (B) actually there is a 100% chance that 200 people will live, and in option (A) there is a 50% chance that everyone will die and a 50% chance that everyone will live. Does this change your decision?

(3) You travel to a nearby city to see a much anticipated concert. The tickets cost $$\$$20 and, among other money in your wallet, you specifically put a $$\$$20 bill in to make correct change. When you get to the ticket window, you pull out the wallet and discover that the $$\$$20 bill is missing. Do you (a) use your other money to buy a ticket anyway? or (b) just go home; you've lost the $$\$$20?

(4) Same as above, except in this case let's assume you bought your $$\$$20 ticket online ahead of time and printed it out and put it in your pocket. You get to the entrance and pull out your wallet and discover you've lost the ticket! The ticket collector is nice about it though and offers to sell you another one. Do you (a) buy an additional ticket? or (b) just go home, cut your losses on the missing ticket?

Prospect theory helps to account for the very weird and consistently non-utilitarian answers that people empirically give to the above questions. It makes it clear that behavioral psychology is needed for the descriptive portion of many different normative mathematical models. At least if you want them to work in practice.

• These are great questions; thanks for the examples! – Christopher Schmidt May 2 '12 at 8:42