this is from Dr. Bob's excellent site:
http://www.drbobsports.com/essays.cfm?p=16
(this is one of an integrated set of really well-presented articles on fundamental topics...check them all out by clicking "essays" at the top of the linked page)
Anyhow here's what's been driving me nuts for the last hour:
-->Marquette has been given a 2% chance to win (paying out 50 to 1), but you believe that their true chance is 3%.
-->North Carolina has been given an 28% chance to win an NCAA Basketball National Title (paying out 3.57 to 1), but you believe that their true chance is 35%.
Which of these is a more attractive bet?
yada yada yada...
then...
If we do a Kelly calculation, we find that we should optimally wager (50*.03-.97)/50 = 1.06% of our bankroll on Marquette. We should optimally wager (3.57*.35-.65)/3.57 = 16.79% of our bankroll on UNC.
I have done and re-done this particular example repeatedly (using my approach, which i long ago determined was equivalent mathematically to the standard "formula", but much more intuitively satisfying)...and
would swear that both of these answers are incorrect...
but this article has been posted on this site for years. I agree with the Kelly bet-sizes calculated in every other example in the other articles in this set, just not this one.
I think it should be "to risk 1.02% of our bankroll (to win our expected value of 50%=(3%/2%)-1 of the roll at odds of +4900 or 50.00...by my logic)
on Marquette...and to risk 9.72% (to win our expected value of 25%=(35%/28%)-1 at odds of +257 or 3.57...by my reasoning)
on UNC".
Either there's a "fossilized" error in the logic i've always employed to do these calculations exposed by this particular example...or a rather blatant error in this article on a prominent site.
If I am wrong, and the article's correct...i'll have to go back to the drawing board on something i long ago decided was fine.