Preface: I've only high-school knowledge in Maths, so this is just an experiment I've run on my pc and I can't motivate the result I get. I'll be wrong in some points for sure so please be nice :)
Consider 2 different skilled people betting on sport events. When they bet on a given event, their probability of winning are $P_1$ and $P_2$.
We suppose that player 1 is more skilled and has an advantage on player 2 (player 2 is not skilled at all and bet randomly, so his probability of success is the same of the probability of the event).
Let's suppose $P_1 = P_2 + kP_2$ where $k$ represent the advantage of player 1 for example $k = 0.05$.
In a given amount of trials $n$ there's a chanche that player 2 has made more correct bet then player 1 (they're betting on events with the same probability of course). I've set $n = 10$.
So I'm calculating $P(X_2 > X_1)$ where $X_1$ and $X_2$ are "success counters" variables.
I expected that $P(X_2 > X_1)$ was maximum where $P(X) = 0.5$, where the variance is maximum, and then decrease simmetrically on both sides. Instead my experiment showed that the maximum was between $ 0.3 < P(X) < 0.4 $.
Why this happen? Is there a function that links $n$,$P(X)$ and $k$?
Is it correct to state that: "In a short-term period we can recognize more easily a more skilled bettor if he's betting on high variance events (those with $P(X) = 0.5$) instead if he's betting on low variance events"?
Is it correct to suppose that: "if we have no competence or if we have to take a negative expected value bet we have better probability of result short-term winner by taking bets with high variance $P=0.5$?"