Ask people about statistics, we have lots of dice. We're going to be considering probabilities when rolling the dice, can they work out the chances of rolling different numbers? What about rolling greater than a 3,4,5? What about rolling at least 3 but not more than 5?

Our game is going to involve us moving across various hexagons in a straight line. Each hexagon has a number (x+) on it, this means it will absorb any dice with x or greater and any lower dies will pass through to the next hexagon and be re-rolled, where they might get absorbed again. You can then talk about independence, we keep re-rolling so we have that!

We'll now talk about cancer and our model. One hexagon will be our tumour cell, the other hexagons are healthy body cells. Our dice will be the rays of radiation we're going to try a destroy the tumour cell with. A healthy cell will be destroyed when it's absorbed 3 dice, we'll aim to maximise the number of dice the tumour cell absorbs. We have 6 possible directions to send radiation from and the goal is to destroy the tumour. The dose is going to be the number of dice we roll and we'll also need to choose a direction, later we'll get more complicated and can choose to split our dose between several directions.

This is basically a simulation study. We can calculate with effort the 'best' dose for a given direction, however it's much more complicated with multiple directions. You can work out the expected damage a dice in a given direction causes to different cells if you want, that's fairly easy. The number of destroyed healthy cells is non-trivial.

Compare total damage to tumour vs destroyed healthy cells from different directions. You can also think of variance. Is the variance in both of these the same? What might our concern be with the variance? High variance of destroyed healthy cells means there's a chance this goes very badly.

While this model seems very simple (it is) you can see how hard it is to calculate the best option. It uses simulations to estimate as the algebra is too complicated. We can do similar more complicated simulations on a computer, and run them thousands of times to account for the variance.

Monte Carlo methods use similar ideas of simulating things, in fact there doesn't even need to be any randomness in the original problem.