A race game for two

A race game for two, ready in ten minutes. You need a sheet of paper with twenty squares drawn in a row, two dice, and one counter each.

Serves $2$, ready in $10$ min. You need $20$ squares in a row, $2$ dice, $1$ counter each.

$2$ players, $10$ min, $20$ squares, $2$ dice.

Method: put both counters on square one. In turn, throw both dice and move your counter forward by the total. The first to reach square twenty or beyond wins.

Start on square $1$. In turn, throw both dice and move by the total. The first to reach square $20$ or beyond wins.

Position $p$, dice $a$ and $b$, total $x=a+b$: $p=1$ at the start, then $p+x$; win when $p\ge 20$.

Tip: the totals are not equally likely. A total of seven can be made in six ways out of thirty-six, so one throw in six gives seven. A total of two comes only from two ones, so one throw in thirty-six.

Totals are not equally likely. Seven: $6$ ways out of $36$, so $\frac{6}{36}=\frac{1}{6}$. Two: only $1+1$, so $\frac{1}{36}$.

$P(7)=\frac{6}{36}=\frac{1}{6}$ and $P(2)=\frac{1}{36}$.

Surprise: the average total is seven. Square one to square twenty is nineteen steps, and nineteen divided by seven is a little under three, so most games last about three throws each.

The average total is $7$. Square $1$ to $20$ is $19$ steps, and $19÷7\approx 2.7$, so about $3$ throws each.

Average total $E(x)=7$, and $\frac{19}{7}\approx 2.7$.