TropicalFruit comments on Dice Decision Making

[TropicalFruit]

Your Latex didn’t quite work.

Also, here’s three quick examples for anyone still wondering exactly how this works. Remember that the chance of flipping tails until a given digit in the binary expansion, then flipping heads, is $(\frac{1}{2}{)}^n$ where n is the digit number (1/​2 for the first digit after the decimal, ¹⁄₄ for the second, etc).

$P=\frac{1}{4}$

$P_2=.01$

My chance to land on the 1 with a heads is exactly $\frac{1}{4}$.

$P=\frac{2}{3}$

$P_2=.101010...$

My chance to land on a 1 with my first heads is $\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+...=\frac{2}{3}$

$P=\frac{1}{7}$

$P_2=.001001...$

My chance to land on a 1 with my first heads is $\frac{1}{8}+\frac{1}{64}+...$

The only semi-tough part is doing the base 2 long division to get from your fraction to a binary decimal, but you can just use an online calculator for that. The coolest part is that your expected number of flips is 2, because you stop after one heads.