# Who’s Making the World Series: Simple Binomial Study

As the MLB playoffs began last night with the Houston Astros completing a three hit shutout over the New York Yankees, we decided to work out a generic method for determining the probability for each team in the MLB playoffs to make the world series. The playoffs finalized with the following schedule:

A.L. Wild Card Game: Houston at New York

N.L. Wild Card Game: Chicago at Pittsburgh

A.L. Divisional Series: Houston/New York vs. Kansas City

A.L. Divisional Series: Texas vs. Toronto

N.L. Divisional Series: Chicago/Pittsburgh vs. St. Louis

N.L. Divisional Series:  New York vs. Los Angeles

Using the continuity corrected probabilities for winning each game, we have the following weighting between each team. Then we walk through all possible scenarios of match-ups and write out the probabilities. For example, the probability the New York Yankees make the World Series is given by

P(Yankees Make the World Series) = P(Yankees defeat Astros) x P(Yankees defeat Royals) x [ P(Yankees defeat Rangers) x P(Rangers defeat Blue Jays) + P(Yankees defeat Blue Jays) x P(Blue Jays defeat Rangers)].

That’s a lot to break down. First, let’s look at the Astros-Yankees game. This is a one game play-off (albeit already played, but let’s assume not yet played) where the Astros have a 54.5% chance of winning the game. If the Yankees are fortunate enough to reverse time and win the game, their next opponent is the Kansas City Royals in the Divisional Series. This series is a best of five series where the Royals have a 40.0% probability of winning each game.

For this series, in order for the Yankees to advance, the Yankees must win three games with the requirement that the Yankees win their third game last. Removing the last game, the first few games follow a binomial distribution, Binomial(N,p,x), with N games played, each with p = 0.4 chance of winning each game. The value x is the binomial probability we want to calculate. Then we have

P(Yankees defeat Royals) = P(Yankees win in 3 games) + P(Yankees win in 4 games) + P(Yankees win in 5 games).

Here, P(Yankees sweep) = Binomial(2,.4,0) x .6 = .216. Similarly, P(Yankees win in 4 games) = Binomial(3,.4,1) x .6 = .2592. Furthermore, P(Yankees win in 5 games) – Binomial(4,.4,2) x .6 = .20736. This means the Yankees has a 68.256% chance of defeating the Royals in a five game series.

Carrying out similar calculations for the Rangers-Blue Jays series (Rangers have a 31.744% chance of winning their series) We can calculate the probabilities of the Yankees winning the Championship series against the Rangers (22.355%) and against the Blue Jays (19.6562%). The resulting probability for the Yankees to make the World Series is then:

P(Yankees make the World Series) = .455 x .68256 x [.22355 x .31744 + .196562 x .68256] = .063706.

This shows that the Yankees have a 6.3706% chance of making the World Series. It’s the lowest probability amongst all MLB teams in the playoffs. Applying the same logic for each game, we obtain the probability for each team to make the World Series.

Probability of making the World Series for each American League team (before Wild Card Game).

For the National League, the second and third best teams in the National League (Pittsburgh Pirates and Chicago Cubs) are forced to play in the wild card game; currently with a third bean ball being sorted out on the field.

For the Wild Card game, the Chicago Cubs have a 56.5% chance of defeating the Pittsburgh Pirates in Pittsburgh. This is holding relatively secure as the Cubs currently are holding on to a 4-0 lead heading into the bottom of the seventh inning.

Filling the probabilities for the Cubs: 37.9491% chance of defeating the St. Louis Cardinals in the Division Series, 40.2355% chance of defeating the Los Angeles Dodgers in the Championship Series, and a 97.5960% chance of defeating the New York Mets in the Championship Series; the Cubs have a 15.8085% chance of making the World Series.

Filling in the probabilities for each other National League team, we have that the Cardinals are the favorite with the Cubs and Pirates right behind them.

Probabilities of making the World Series for each National League team.

Wait… Cubs are likely to beat the Pirates; but the Pirates have a better chance?

We take a quick note to identify that while the Cubs are more likely to win the Wild Card game, the Pirates have a better chance of making the World Series. This is in part due to the fact that the Cubs are unlikely to defeat the Cardinals and Dodgers compared to the Pirates (45.8803% chance over the Cardinals, 87.3964% chance over the Dodgers). However, if the Pirates are unable to come back against the Cubs tonight, it’s all for naught.

Houston Eliminated the Yankees. Now What?

Now we update the playoff probabilities as the Yankees’ slim 6 percent chance gets reshuffled into the mix. However, these percentage points don’t just go to another team. In fact, the elimination of the Yankees is a blessing for the Kansas City Royals as the Royals had a 31.744% chance of defeating the Yankees. Now that the Yankees are gone, the Royals are guaranteed to play the Astros and have a 68.256% chance of winning the series. Making this reshuffle, the probability of making the World Series for each team is updated, showing the Royals leap-frogging the Texas Rangers.

Updated Probabilities of making the World Series for the American League teams after Houston’s elimination of New York.

Remember, these are merely probabilities. This is the MLB post-season and anything is possible. For now, we should expect to see a Blue Jays – Cardinals World Series based on the trend of the regular season.

Update: Chicago Cubs Defeated Pittsburgh Pirates 4-0

With the Chicago Cubs advancing to the Divisional Series against the St. Louis Cardinals, the probabilities for each of the National League teams adjust to recoup the probability originally associated with the Pittsburgh Pirates.

Updated probabilities of making the World Series for each of the remaining National League teams.