Copy Cards

What is it?

This is a group project I had to work on where we had to create a casino game and calculate all the juicy maths behind it.

The Game

The played draws a card out of a closed box. The card gets noted a put back. The player does this a total of 10 times. If the same card gets drawn twice the player loses, and if that doesn't happen, the player wins. After 3 cards drawn, the player can decide to double their bet. When the player wins, they get back double their bet, and when they lose, the casino keeps the money.

Chances

Birthday Paradox

This game is based on the Birthday paradox. It sounds intuitive that you need quite a lot of people to have a 50% chance of 2 people sharing a birthday. However, you only need 23 people. The formula to calculate this is: $P(c)=1-\frac{n!}{(n-k)!\cdot n^{k}}$. In this formula $n$ is the amount of birthdays (normally 365), and $k$ is the amount of people.

Expected value

With the previous formula we can calculate the expected value. $$P(c)=1-\frac{52!}{(52-10)!\cdot52^{10}}\approx60.2\%$$ After 3 cards drawn, the played can double their bet. These are the chances the casino wins before the doubling, after the doubling and the player winning. $$P(c_{1-3})=1-\frac{52!}{(52-3)!\cdot52^{3}}\approx5.7\%$$ $$P(c_{4-10})=1-((1-\frac{52!}{(52-3)!\cdot52^{3}})+\frac{52!}{(52-10)!\cdot52^{10}})\approx54.6\%$$ $$P(p)=\frac{52!}{(52-10)!\cdot52^{10}}\approx39.7\%$$

Money

Therefore, the casino approximately earns: $$m(g,s,d)=g\cdot(s\cdot0.056+s(b+1)\cdot0.547-s(b+1)\cdot.397)$$ Where $m$ is money, $g$ is the amount of games, $s$ is the bet and $d$ is whether the player doubles down (this is a boolean value, 0 or 1). Imagine the game get played $10 000$ times, and the player doubles down every time he draws his third card. His initial bet is €10. $$m(10000,10,1)=10000\cdot3.56=€35600$$

Normal distribution

Let's play the game 10000 times again. When $n$ gets closer to infinity, we can assume that:

$$\mathcal{B}(n,p)\sim\mathcal{N}(np,\sqrt{np(1-p})$$ Now we can calculate the values of $\mu$ and $\sigma$.

$$\mu=np=10000\cdot0.602=6020$$

$$\sigma=\sqrt{np(1-p)}=\sqrt{10000\cdot0.602\cdot(1-0.602)}\approx48.9$$ We can use this to plot a graph.

We can see that there is a 65.9% chance of the casuno winning 6000 or more of the 10000 games.

Installation

To play the game you have to do one of these things:

Go to releases and download the file corresponding to your OS.

You can also install the game with cargo. Beware that you do need Rust installed.

cargo install copy-cards

If that doesn't work, or you want to build locally, here's what to do. You need Rust installed for this too.

git clone https://github.com/7ijme/copy-cards
cd copy-cards
cargo build --release

Now you can play the game, have fun!