Note: This is the first of a two-part series on Probability. The baseball odds used in this article were true on the morning of October 11 and are definitely no longer true. Using this gambling advice will likely cost you money in the long run, as Joe will explain in Part 2.
In terms of excitement, the World Series of Coin Flipping would rate right next to solving Data Sufficiency problems in the general public’s mind. But any Vegas oddsmaker worth his weight in comped beverages would be able to calculate the odds for every coin flipping contest in the time it would take the coin to land. In an 8-team bracket-style tournament, every squad would have 1 in 8 odds to win it all because only 1 team out of the 8 could outguess their way to the championship. But what if each matchup of two teams was a 3 game series? It wouldn’t affect the odds at the beginning of each series (still a 50% probability for each team to win), but once the first outcome was decided, those Vegas oddsmakers would require some knowledge of “And Probability” to keep the odds fair.
Imagine we have two teams competing in the first round of our Coin Flipping Playoffs- let’s call them Baltimore & New York. In our three game series, New York happened to have some late flipping heroics to go up 1-0. What are the odds that Baltimore comes back in this series and what are the odds that Baltimore comes back and then wins the whole tournament? The key to this type of question is understanding that many things have to go right in order for Baltimore to win it all- first they need to win Game 2, then win the winner-takes-all Game 3, then win their semi-final series, and then still have enough thumb strength to flip the World Series in their favor. Four unique events need to happen, and every single one of those events must happen for Baltimore to emerge victorious. Since the odds of each game and each series going Baltimore’s way would be 1/2, we can solve this by finding the odds that Baltimore wins Game 2 (1/2) AND Game 3 (1/2) AND the next series (1/2) AND the championship (1/2). In probability, whenever we want X AND Y to occur, we need to multiply the respective odds together. The odds of Baltimore winning Game 2 and 3 would be 1/2 x 1/2 = 1/4. The odds of winning all four events would be (1/2)4 = 1/16.
New York’s probability to win it all would require a slightly different calculation. A soon-to-be-fired oddsmaker might look at the potential for New York to emerge victorious from the first round of the playoffs and say, “New York could win after Round 2 AND it could win after Round 3,” and set the odds at 1/2 x 1/2 = 1/4. His fellow oddsmaker might look at the same situation and think, “There are exactly three outcomes here and New York would win two of them: New York could win in Game 2, New York could win in Game 3, or Baltimore could win in Game 3,” and she would be joining her co-worker in the unemployment line after setting the odds of New York advancing past the first round at 2/3. But at least she tried to apply the second major calculation concept within probability- “Or Probability.”
A fair six-sided dice should have a 1 in 6 probability of landing on each side. Which means that the odds of landing on a 2 (1/6) OR a 3 (1/6) would require you to add 1/6 + 1/6 = 2/6 = 1/3. But while there are three different outcomes that could occur in the Baltimore-New York series, not every event is equally likely. Let’s take a look at the odds of each event and combine our knowledge of AND & OR probability:
1) New York wins in Game 2- 1/2 probability that they win Game 2. Series over.
2) New York wins in Game 3- in order to get to a Game 3, Baltimore would need to win Game 2 (1/2 probability) AND New York would need to win Game 3 (1/2 probability) = 1/2 x 1/2 = 1/4
3) Baltimore wins in Game 3- in order to get to a Game 3, Baltimore would need to win Game 2 (1/2 probability) AND Baltimore would need to win Game 3 (1/2 probability) = 1/2 x 1/2 = 1/4
Since New York has two ways to win the series (winning in Game 2 OR losing Game 2 but then winning Game 3), New York has a 1/2 + 1/4 = 3/4 probability of advancing to the next round.
But like the GMAT, life is all about shortcuts. The fastest oddsmaker was able to come to the same answer by using something called the “1-x Trick.” Probability is about finding the certainties in life, and divvying up the likelihood of how different events could play out. If we eliminate the probability that uncertainties occur (coin flipping becomes outlawed, the players decide to all retire at once, someone catches the coin over the right field wall), we can with 100% certainty state that either New York OR Baltimore will win the series. Which can be expressed as NY + B = 100% or NY + B = 1. In order to calculate the odds of New York beating Baltimore, we had to account for two different scenarios, carefully calculate the odds of each event occurring, and then add the final probabilities. But since it was much easier to calculate the probability of Baltimore winning the series (1/4), we could have inserted that probability into our equation:
NY + B = 1 —> NY = 1 – B —> NY = 1 – 1/4 = 3/4
This “1-x Trick” is a big timesaver on the GMAT when there are several different ways that something could happen, but only one way when something could NOT happen (or vice versa).
One thing to note about AND vs OR probability is that since all odds of an event fall between 0 and 1 (or 0% and 100%), you can often stop and think about whether you want to increase or decrease the odds of something occurring to help you decide whether you need to multiply (which would reduce the odds) or add (which would increase the odds). What are the odds of me becoming a professional baseball player OR becoming the General Manager of a team? While neither scenario has great odds, I can take solace in the fact that I have slightly better (and not worse) odds that one of the two things could happen (any baseball owners out there???). But the odds of me doing both of these things, a player AND a manager, looks worse than my high school batting average.
So what does all of this have to do with baseball playoffs? As I write this article, the Yankees of New York are playing the Orioles of Baltimore, and after splitting their first two games, New York happened to win their first of a de-facto three game series to advance to the ALCS. And Las Vegas decided that these next two games would not be decided by a coin flip. By one sports book, the Yankees are the favorites to win the World Series by winning one of their next two games against Baltimore, then the American League pennant, and then the World Series at 3 to 1 odds. This is Vegas’s way of saying that they think there is a (1/(1+3)) 25% probability that they think the Yankees will win it all (we’ll talk more about how to convert these types of odds next week). But IF every baseball game were determined by a coin flip, the odds that the Yankees win one of the next two games (3/4) AND the pennant (1/2) AND the World Series (1/2) is only 3/4 x 1/2 x 1/2 = 18.75%. The Orioles on the other hand are a 20 to 1 long shot (~4.8% odds) to win it all even though if they had a 50/50 chance of winning the next two games (1/2 x 1/2 = 1/4) AND the next two series (1/2 x 1/2 = 1/4), Baltimore would have a 1/4 x 1/4 = 1/16 probability of making Orioles Magic happen. Obviously baseball isn’t a coin flip and when you see certain prognosticators give the Yankees a 57% chance of finishing the series in Game 4, it begins to explain why these lines aren’t so simple. Yet many men and women have lost a pretty penny in Vegas and next week we’ll take a closer look at these odds to see why no matter who wins in October, Vegas wins all year round.