This is essentially a three-step process for each team:
1. Determine the target number of wins to achieve the team's desired outcome.
For example, the goal might be to clinch a division or to win a playoff series. In the case of a division race, the target (magic number) can also be reached by virtue of the other team losing games.
2. Estimate the winning chances for both teams in each relevant remaining game.
I'm a professional baseball handicapper. It's my job to get this step right, although I'm not always going to be 100% accurate.
"Relevant" game means we're dealing with all games involving the team itself, plus the teams chasing them in a division race, and their potential future opponents in the playoffs. For example, the current Red Sox probability to win the World Series is affected by every game played, because each game changes their likelihood of facing a given opponent later on. Their chances to win the AL are affected only by AL games.
It is this step which I hope will separate me from the pack of others who have developed playoff odds estimates. If Clay Davenport ever decides to start handicapping individual games, I'll be out of business in no time.
3. Use probability calculations to determine the chance of the team achieving its goal.
If a team needs to win three games in a row with winning probabilities of .474, .475, and .395, it has a (.474)(.475)(.395) = 8.9% chance to win all three. (This is the actual situation the L.A. Angels are in right now.)
Most calculations are going to be more complex than this, which is why I'm doing them for you.