Question: Consider a dice game in which you try to obtain the largest number (*e.g.* you win a prize based on the final dice roll).

1. You roll an 8-sided die, with numbers 1–8 on the sides.
2. You may either keep the value you rolled, or choose to roll a 12-sided die, with numbers 1–12 on the sides.

What's the best strategy for choosing what to do in step #2? 

I know the 8-sided die has expected payoff of 4.5, and the 12-sided die has expected payoff of 6.5. So I think relying on the 12-sided die is better — but how do I show the probability of this?

Answer: 
If you are trying to maximise the expected score, then since the expected value of the 12-sided die is $6.5$, it makes sense to stop when the 8-sided die shows greater than $6.5$, i.e. when it shows $7$ or $8$, each with probability $\frac18$. So with probability $\frac34$ you throw the 12-sided die.

The expected score is then $$7 \times \frac18 + 8 \times \frac18 + 6.5 \times \frac34 = 6.75.$$
This is easy enough to break down with game theory.

As the objective is to get the highest number, you only want to roll again when you're more likely to roll a number that's greater than your first score.

```
N | P(>)
=========
1 | 11/12
2 | 10/12
3 |  9/12
4 |  8/12
5 |  7/12
6 |  6/12
7 |  5/12
8 |  4/12

```

As you can see it's a linear regression toward `0`. The important point is that when the first roll is `6` the odds are `50/50` that you'll improve your score.