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 (12)n where n is the digit number (1/2 for the first digit after the decimal, 1⁄4 for the second, etc).
P=14
P2=.01
My chance to land on the 1 with a heads is exactly 14.
P=23
P2=.101010...
My chance to land on a 1 with my first heads is 12+18+132+...=23
P=17
P2=.001001...
My chance to land on a 1 with my first heads is 18+164+...
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.
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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 (12)n where n is the digit number (1/2 for the first digit after the decimal, 1⁄4 for the second, etc).
P=14
P2=.01
My chance to land on the 1 with a heads is exactly 14.
P=23
P2=.101010...
My chance to land on a 1 with my first heads is 12+18+132+...=23
P=17
P2=.001001...
My chance to land on a 1 with my first heads is 18+164+...
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.