If you have X:1 odds, when should you bet?
Generally when you are betting, X:1 odds means that if the event in question were to be repeated X+1 times, the even would NOT happen X times and happen only 1 time. Usually odds are stated so that the odd of not happening is first.
In other words, your probability of winning is 1/(X+1) and your probability of losing is X/(X+1). In such a game, you should bet if the payout is more than $(X+1) * [Your Bet]. Let’s try a concrete example.
Say you are betting that the 49ers win the Superbowl with 3:1 odds. This means you the 49ers have a 1/4 chance of winning and a 3/4 chance of losing. The bookie says that if you bet $10 and you win, then he will pay you $10 * 4/1 = $40. Should you do it?
Your expected value from the bet would be (1/4) * $40 + 3/4 * 0 = 10, so you would be indifferent at this exact amount. For any amount greater than $40 you should take the bet, and for any amount less than $40 you should turn it down. Of course, this is all assuming that your ultimate goal is to maximize your expected amount of money.