Coin Toss Game
11-29-2017
, 05:33 PM
11-30-2017
, 02:25 AM
As far as I can determine, the formula above gives identical results to Nick's formula previously posted in this thread.
(For large number of rounds, Nick's formula is recommended as it does not encounter any overflows.)
To repeat: very very good stuff in this thread. Kudos to all.
11-30-2017
, 09:54 AM
Quote:
I have not looked into this, but many formulas can give the correct result for small inputs.
The lead change question only becomes interesting (i.e., tricky) when the number of rounds leaves the range of "small" numbers.
My gut tells me that your formula is incorrect for number of rounds greater than 4 (or something like that). It would be hard to believe that the disparity in results is due to lack of precision.
Edit to add:
The following results are known to be correct (here I give them as fractions to help avoid any precision problems).
E(1) = 2 / 4
E(2) = 12 / 16
E(3) = 58 / 64
E(4) = 260 / 256
E(5) = 1,126 / 1,024
E(6) = 4,788 / 4,096
E(7) = 20,140 / 16,384
E(8) = 84,120 / 65,536
E(9) = 349,606 / 262,144
E(10) = 1,447,556 / 1,048,576
The lead change question only becomes interesting (i.e., tricky) when the number of rounds leaves the range of "small" numbers.
My gut tells me that your formula is incorrect for number of rounds greater than 4 (or something like that). It would be hard to believe that the disparity in results is due to lack of precision.
Edit to add:
The following results are known to be correct (here I give them as fractions to help avoid any precision problems).
E(1) = 2 / 4
E(2) = 12 / 16
E(3) = 58 / 64
E(4) = 260 / 256
E(5) = 1,126 / 1,024
E(6) = 4,788 / 4,096
E(7) = 20,140 / 16,384
E(8) = 84,120 / 65,536
E(9) = 349,606 / 262,144
E(10) = 1,447,556 / 1,048,576