Assume we analyze bets on a game with 3 possible outcomes: win, lose, draw. Let's review bets for teams A and B with 3 possible outcomes:
- Number of bets that A wins = $N_{A}$;
- Number of bets that B wins = $N_{B}$;
- Number of bets on draw = $N_{A,B}$.
Is there any statistical bias towards growing probability for draw when $N_{A}=N_{B}$ ?
EDIT (Trying to add more background)
;TLTR; (I try find confirmation that $max(P_{N_{A,B}})$ is when $N_{A}=N_{B}$ )
Let's start from very simple case, where we have fixed audience = 10000 who make the bets. Fixing the audience makes our N's dependant of each other. Let's spin random number generator and evaluate $N_A, N_B$ and $N_{A,B}$ as following:
na = random.randrange(0, audience)
nb = random.randrange(0, audience - na) # random of the rest of audience
nab = audience - (na + nb) # the rest of audience is draw
Following hystogram is obvious (nab lays on Z-axis). The max of draws is close to reciprocal minimum of $N_A, N_B$. And "plato" of relativly big number of draws has strong edge near 5000.
But now Let's make step back to complexity. What if our audience uses some insights (hidden variables) - like a prior knowledge about teams A and B. Or any participant of audience can make more than 1 bet simultanously.
Can we state then that equality $N_{A}=N_{B}$ also means increase of probability for draw outcome?
EDIT 2 I went even further, just created script where color (magenta, yellow and silver) plays role of hidden variable that boosts one of $N_{A,B}, N_{A}, N_{B}$. Probability of draw has well visible tendency when $N_{A}, N_{B}$ are close to each other. But how can it be prooven?
