## Betting Math

Leslie Green’s Casino Problem

A bunch of Physicists, fresh from College, go to their local casino with plans to profit with their advanced gaming strategies for Roulette. They have chosen a Roulette game where there are 36 numbers from 1 to 36, alternately colored black and red. There is only one green slot, which is usual in Europe but unusual in the USA. When the ball lands in the green slot, 0, all bets are lost to the House.

All participants place 223 bets. Each bet pays out at 1/1 odds, meaning you get your bet back plus an equal amount of winnings.

John bets only on RED, and only immediately after a run of 3 reds.

Sally bets only on EVEN, but only after the sequence ODD, ODD, EVEN, ODD.

Peter looks for a run of 8 events, either all ODD, all EVEN, all RED or all BLACK. He then bets against this run continuing.

At the end of the 223 bets, who is the most likely to have won?

Leslie Green’s Sports Betting Problem

The all-male under-10 football team, the Hunky Heroes, are playing the all-girl under-12 football team, the Girly Girls. I am facilitating the betting, although this may not be strictly legal in all jurisdictions.

Buddy and his friends have put a total of \$200 on the favorites, the Hunky Heroes, to win. If they win I have to pay out a total of \$3 for every \$2 placed.

Samantha and her friends have put a total of \$50 on the Girly Girls to win. If they win I will pay out a total of \$6 for each dollar placed. (Some people would call this 5/1 odds, where you win \$5 for each dollar placed AND you get your \$1 bet back).

Some of the parents have individually placed a total of \$100 on a draw. In this case I pay out a total of \$3 for each dollar placed.

Who has the greatest chance of winning?

Casino Statistics

In a casino on a Sunday,  90% of the visitors lost \$200 each, # 9% of the visitors lost \$1,000 each, and # the rest won \$10,000 each. # If the profit of the casino is \$340,000, how many people visited the casino?

Jane and Gerry visit a casino. In one game, they have a 1/5 probability of winning \$100 and 1/2 probability of losing \$50. They have also a chance of no win / no loss. What is the most likely amount of money they will win (or lose) at the end of 100 games?

## Coin Landing On Its Edge

I flip a fair Swiss franc* and it falls in mousse**. What is the probability that the coin stays on its edge after the mousse melts?

*One Swiss franc is about 1 USD; diameter 23.20mm , thickness 1.55mm, weight 4.4g.  During a coin toss, the coin is thrown into the air such that it rotates edge-over-edge several times.
**A mousse (French ‘foam’) is a prepared food that incorporates air bubbles to give it a light and airy texture.

Solution

The probability  that the coin stays on its edge is about 4% for a Swiss franc and 5% for an American dime.

http://www.aplusclick.com/t.htm?q=5135

Such an outcome on a hard and flat surface is fairly unlikely, having been estimated at approximately 1 in 6000 tosses.

https://en.wikipedia.org/wiki/Coin_flipping