How to use your opponent's ignorance to win at dice.

I found this true story in Against The Gods (The Remarkable Story of Risk) by Peter L. Bernstein.

Chevalier de Mere (real name Antoine Gombaud) was a Frenchman who, in the 1600s, made money with this bet:

"I bet I can get a 6 within 4 throws of the dice."

The book does not explain the calculation, but does give the answer to his chance of winning (roughly 51.8% of the time). I'd like to be good at probability calculations, but I'm not. Anyway I thought I'd try to find out how the probability was calculated. I preceded like this:

...which yields...

I was very proud of myself until I found out that there is a much easier way of calculating it. Just count out all the ways of losing, and take that number away from all the possible outcomes (winning + losing) and you've got the answer.

To lose you must throw a non six four times. The probability of throwing a non-six is 5/6. To do it four times you've got (5/6)*(5/6)*(5/6)*(5/6). And so the simpler way of calculating the answer becomes:

I wondered how many games and with how much money I'd have to play (betting on the ignorance of my opponents) so I could give up doing a real job. This would presumably be illegal, so the income would be tax free. Let's pretend I'd be happy with 30,000 tax free Euro a year.

1.774% x Investment = 30,000
Investment = 30,000/0.01774 = 1,691,093 Euros. Per year.

I'd somehow have to pursuade people to play the game with me with a stake of roughly 4500 Euros a day. I think I'll keep the day job.

As I say I'm no probability genius and will be happy to be corrected.