# ETOOBUSY ðŸš€ minimal blogging for the impatient

# Double check a puzzle result

**TL;DR**

I double checked a puzzle (theoretical) solution with some simulation in Raku.

I have a free account on Brilliant, which means I can enjoy a little puzzle a day from them.

These puzzles are normally meant to be solved theoretically, that is by thinking a bit on the problem and solving it e.g. with a couple of formulas.

One recent puzzle was a interesting question about a game similar to this:

Roll a six-sided die and accumulate as many points as the outcome. If the outcome is greater than 2, repeat to gain more points; if the outcome is 1 or 2 stop.

What is the expected value of the points accumulated?

Assuming a fair six-sided die, each face value has probability
$\frac{1}{6}$ of coming out, so the expected value *for a single round*
is:

Then, of course, we have to consider that we would have to go on if the outcome is greater than $2$. If we call our target, unknown expected value $E$, this additional component that we have from continuing will have to be considered only $4$ times out of $6$, i.e. only when the outcome is one of $3$, $4$, $5$, or $6$. It contribution, then, will be $\frac{4}{6} E = \frac{2}{3} E$.

Overall, then, our $E$ will be formed by the outcome of a single roll $E_1$ and this additional component, i.e.:

\[E = E_1 + \frac{2}{3} E = 3.5 + \frac{2}{3} E\]We can now solve for $E$:

\[E - \frac{2}{3} E = 3.5 \\ \frac{1}{3} E = 3.5 \\ E = 10.5\]On average, then, our score will be 10.5 points.

Did I get the calculations right? Letâ€™s set up a simulation and double check!

In pure *bottom-up* style, letâ€™s start defining one full round of the
game:

```
sub simulation-round () {
return [+] gather {
loop {
my $value = roll-die();
take $value;
last if $value < 3;
}
}
}
```

The `loop`

goes on *indefinitely*, although it has a positive and
finite probability of being interrupted thanks to statement ```
last if
$value < 3
```

, that is exactly our exit condition for a round of the game.

We use `gather`

/`take`

to get the outcome of each roll of the die;
as our score is actually the *sum* of all these outcomes, we use the
`[+]`

reduction operator to obtain this sum.

Now, to *estimate* the expected value we want to double check, we can
set up *a lot* of simulation rounds and take the average of the
outcomes:

```
my $N = @*ARGS.shift || 100;
my $total = [+] gather { take simulation-round() for 1 .. $N };
put 'average gain: ', $total / $N;
```

Again, we calculate the total sum of *all* the `$N`

outcomes using
`[+]`

, applied to a `gather`

/`take`

pair that operates on whole
simulation rounds this time. At this point, the average is calculated by
dividing this `$total`

by the number of rounds `$N`

that we did.

The die rolling will be done a bit crudely, leveraging the stock
`rand`

facility. Iâ€™m not sure about its statistical characteristics,
but for our simulation it will do:

```
sub roll-die (Int:D $sides where * > 0 = 6) { (1 .. $sides).pick }
```

Iâ€™m also not entirely convinced that putting the default value *after*
the condition makes this entirely readable, but itâ€™s life.

Letâ€™s run this a few time, by averaging over 10000 rounds each time:

```
$ for i in 1 2 3 4 5 ; do raku multiple-rolls.raku 10000 ; done
average gain: 10.5096
average gain: 10.4038
average gain: 10.3704
average gain: 10.3157
average gain: 10.6384
```

It seems pretty consistent with the theoretical value of $10.5$ that we calculated above, yay!

If you want to play with the code, there is a local copy here.

Have fun and stay safe!