Probability

UK Secondary (7-11)

Can you win at the casino? (Investigation)

Lesson

Casinos are big business. In 2014 the US spent around $20 billion on chocolate, but they spent more than $60 billion on gambling, more than three times as much! Chocolate is delicious, but what is it about gambling that makes people want to spend their money on it?

Most people gamble for one simple reason: because they think they will win more money.

__Option #1: Be good at maths, and stay away from the casino__

However, people who know maths stay away from casinos. This is because they know that all of the games within the casino slightly biased, so that people who play them lose more money than they win. How do they do this? Let’s take a look at a particular example,

Roulette is played by spinning a ball on a wheel with $38$38 numbers on it. You pick a number, then they spin the wheel, and if it lands on your number, you win! So, what’s the chance of you winning?

Well, there are $38$38 different numbers, and the ball is equally likely to land on any of them. Probabilities all have to add up to $1$1, because in probability $1$1 means certainty, and we can be certain that something has to happen. So if we have $38$38 equal probabilities, which all have to add up to 1, the only way this can happen is for each probability to be $\frac{1}{38}$138., that's only $2.6%$2.6%

So, the people at the casino have worked out that the probability of you winning is $\frac{1}{38}$138. They now need to figure out how they can use this to make money from you. The way they do this is by setting the winning amount: they make it so that when you bet $\$1$$1 and get it right, you get $\$35$$35, instead of $\$38$$38 as you might expect.

However, there is a twist to this story. The Casino’s calculations are based on abstract theory, and like all abstractions, it may not work perfectly in the real world. The key assumption under this model is that the probabilities of the ball landing on any number are equal. Is this really true in the real world?

__Option #2: Be really good at maths, and go to the casino__

Some people decided to see if they could defeat the casinos mathematical model. They reasoned that the roulette wheel might not be perfectly balanced, and therefore that some numbers might be more likely than others. If this were true, then the Casino’s mathematical model would break down, and it would be possible to make more money than you lost.

Joseph Jagger and his team of six accomplices were the first to use this method in 1873, by watching all the wheels in the casino for a while and calculating the chances based on what they saw.

For example, let’s say they saw a wheel being spun $1000$1000 times, and landing on the number $7$7, $40$40 times. This means the chance is somewhere around $\frac{40}{1000}$401000, which is equal to $\frac{1}{25}$125, a larger probability than $\frac{1}{38}$138.

By continually betting on the number 7, and winning roughly every $25$25 times instead of every $38$38 times, you could make more money than you lost. Jagger and his team won $\$325000$$325000 this way!

Later Casinos tried to prevent these kinds of problems by fixing their wheels to make them more balanced. However, small variations still existed, and the fact that they allowed people to make bets while the ball wall still rolling made it even easier to use maths to make prediction about where it would land. Although the maths required was much too complex to do in one’s head, in the 1970’s a group of university physics students called the Eudaemons made special computers that would fit into their shoes to do the maths for them. They would input the numbers into the computer by tapping their feet, and receive the answer as a vibration!

__Option #3: Be really really good at maths, and own the casino__

Despite the use of sophisticated techniques to try to undermine the mathematics of casinos, the casinos eventually notice them. They use even more complex mathematics to spot unusual patterns of winning in their casinos, and then use surveillance to catch those people in the act of cheating, which can result in those people getting arrested (and having to give the money back too).

In the end, the only sure way to make money at the casino is to own it!

**Questions:**

Imagine a roulette wheel with $100$100 slots instead of $38$38. What is the chance of winning?

- What should the casino set the winning payment as in order to make money?

Imagine you used a sophisticated computer in your fingernail to count the number of times the wheel lands on each number, and found that a particular number came up $3$3 times after $100$100 throws.

- What is the approximate chance of getting the same number on a single spin?
- If you bet on this number $100$100 times, how much money would you expect to make?

Imagine that you are the casino owner, and you have kept track of the money made and lost by every customer. Of $10000$10000 people who have played roulette, $300$300 have made more money than they lost.

- What is the approximate chance of a particular person walking away from the roulette table with more money?
- When you see this young man/woman with a strange looking fingernail winning huge amounts of money every day for a week, what do you do?