Suppose that a betting place is willing to pay $\$11$ dollars for each $\$10$ to anyone that bets on a certain football team (say A) that plays next saturday. Does that mean that it's willing to pay $\$19$ for each $\$10$ to anyone that bets on the opposing team? My reasoning is that a betting place, if it knew team A is going to win for sure, would double the pay (so anyone that bets $\$10$ would receive $\$20$). In this case, the betting place is only willing to pay $$\$1$, so the other $\$9$ should go to the opposing team. Is this correct?
Another interesting fact is that, from the betting place perspective, team A seems to have a higher probability of winning the match. How could I compute this probability? (from the betting place perspective).
No, the betting place needs some return for expenses and profit. If you bet both sides of a match you are guaranteed to lose money. They are really functioning as a broker for bets. Their objective is to keep the amount of money bet each way to the proper proportion so they make the same money whatever the result.
If you want to compute the odds on the assumption that 11-10 is fair, note that if the first team wins 10 games and the second wins 11 games the payoff is zero, so these odds say the first team should have $\frac {10}{21}$ chance of winning.