# Dice game of probability (Just for fun)

(This will never be in an exam but adds fun to maths and probability and teaches you the language and the strange nature of probability. Teachers should try this in class for fun)

Look at these unusual dice:

We will open each up.

The first is:

The second is:

The third is:

The fourth is:

If we rank the dice as which is stronger, which is weaker by the definition:

"one die is **stronger** than the other die if that die has a **higher** probability of winning against the other die"

which means it wins more frequently.

Conversely

"One die is **weaker** than the other die if that die has a **lower** probability of winning against the other die."

Which means it loses more often.

**Example 1**

Which is the stronger dice?

So the totem pole ranking would be:

**Example 2**

Which is the stronger dice?

So the totem pole ranking would be:

**Example 3**

Which is the stronger dice?

Half the time this comes up `1` and if it did it

would lose every time against `6`, `6`, `2`, `2`, `2`, `2`

**LOSE**

**LOSE**

**LOSE**

**LOSE**

**LOSE**

**LOSE**

Half the time this comes up `5` and if it did it

would lose twice against `6`, `6`, `2`, `2`, `2`, `2`

**WIN**

**LOSE**

**WIN**

**LOSE**

**WIN**

**WIN**

This loses 50% of the time `+` a bit more loss

So this is a weaker die

So the totem pole ranking would be:

**Example 4**

Now lets compare the two ends. Which is stronger?

You would think this would be the `6`, `2`, `2`, `2`, `2`, `6` because of the above rankings but

`3`, `3`, `3`, `3`, `3`, `3` wins four out of six times. So this is the stronger die.

So the new totem pole ranking would be:

**Example 5**

If we do this again and compare the two ends we are comparing:

and we already know the answer (see example 1)

AND SO THE CYCLE CONTINUES

**NOTES ABOUT THIS EXAMPLE**

This cycle is called a non-transitive cycle in probabilistic comparisons.

This is important for people to know because if you compare `3` drugs:

If A cures more people than B in one sample and B cures more people than C. You can not necessarily conclude that A is better than C. It might be true but not necessarily.

Half the time this comes up `5` and if it did it would win every time against `4`, `4`, `4`, `4`, `0`, `0`

**WIN**

**WIN**

**WIN**

**WIN**

**WIN**

**WIN**

Half the time this comes up `1` and if it did it would win twice only against `4`, `4`, `4`, `4`, `0`, `0`

**WIN**

**LOSE**

**WIN**

**LOSE**

**LOSE**

**LOSE**

This wins 50% of the time `+` a bit more.

So this is a stronger die.