Dog racing (and other animal racing) is one of the most widespread gambling genres in the Empire. This is a basic mechanic for gambling at a race track:
The key to success when gambling at the races is to discern which animals have a better chance of winning their race than the posted odds would indicate. Rather than posting up the complete payoff odds for every dog in each race, this gambling mechanic takes a shortcut--you ask around the track for a while, and then the GMs make a secret roll (using your Int with Gambling skill) to determine how many "good bets" you turn up. However, betting at the track is inherently Perilous. For each "1" you roll, you'll get a "good bet" which is actually a "bad bet"--the named animal actually has a much poorer chance to win his race than the odds indicate. At some tracks, the peril number may be higher (i.e. it could be that all 1s and 2s are bad bets, and so on). This is why the roll is secret. All you will see is a set of bets that you think are good.
A typical "good bet" is described as follows: "Dog #1 has 4 dice and 5 skill, paying off 3-1" (i.e. you bet 1, and if you win, you get your 1 back plus 3 more.)
This doesn't explicitly state the number of successes needed to win the race. Sometimes the GMs may give you that; in this case, by running the odds of making X successes with 4 dice and 5 skill, you can pretty much figure out that the target must be 4 successes. 4 dice and 5 skill has about a 27% chance of making 4 successes, so at a 3-1 payoff, this is a good bet. (Unless it's actually a bad bet, of course.)
The mechanic described here is a pre-packaged mechanic that assumes that you're doing all of the clever things you can do; it's fairly friendly to skilled characters as long as you can endure a high variance in outcomes (some days, none of your 4-1 bets come in...). So if you want to do crazier stuff (in puttering, at least), you probably don't want this mechanic--you want to just go to the races and start from scratch.