3rd at Pro Tour Oath of the Gatewatch, 7th at Pro Tour Shadows over Innistrad, and 4th at Pro Tour Eldritch Moon: Luis Scott-Vargas put up an absurd streak over the course of the last 3 Pro Tours. The last time someone made 3 consecutive Top 8s was 18 years ago, achieved by none other than Jon Finkel.
But how likely is it, really? Is it truly extraordinary for someone of Luis’ skill level to make the Top 8 of the last 3 Pro Tours?
Before I do the math, I encourage you to take a guess what the likelihood is. You may assume that Luis’ probability of winning a match this season is 62.5%, which is his lifetime win percentage at the Pro Tour (according to a graphic shown during the Top 8 stream).
While you think about this, let me help build your order-of-magnitude intuition via the following list of probabilities.
- 50% (1 in 2): The probability of a fair coin toss turning up heads.
- 16.7% (1 in 6): The probability of rolling 1 on a fair six-sided dice.
- 2.8% (1 in 36): The probability of rolling snake eyes, i.e., two dice both showing 1.
- 0.3% (1 in 365): The probability that your parents have the same birthday.
- 0.07% (1 in 1440): The probability of being born at the exact minute of 11:30 a.m.
- 0.01% (1 in 8192): The probability of 13 coin tosses all coming up heads.
- 0.002% (1 in 50,000): The probability of the earth being hit by a 300-meter asteroid in a given year.
- 0.0001% (1 in 730,000): The probability of a giant volcano eruption at Yellowstone in a given year.
- 0.0001% (1 in 960,000): The probability of being struck by lightning in a given year.
- 0.00002% (1 in 4,700,000): The probability of being killed on a single airline flight.
So where do LSV’s Top 8s at the last 3 Pro Tours rank?
To figure out the answer, I can use the model I described two weeks ago. By setting LSV’s overall win percentage to his lifetime average of 62.5%, the model predicts an 11.4% probability of making it to the Top 8. Raising that number to the third power, you find that LSV’s probability of making the Top 8s at the last 3 Pro Tours is approximately 0.15% (1 in 675).
In other words, it’s somewhere in between the probability that your parents have the same birthday and the probability that you are born at the exact minute of 11:30 am.
Weirdly enough, both of those things apply to me, so I certainly picked the wrong things when they were handing out unlikely events. Oh well.
The Law of Truly Large Numbers
It is important to keep in mind that given enough opportunities, any outrageous thing is bound to happen eventually, no matter how unlikely it is at any opportunity. A simple example is a lottery: Even though an individual has a low chance of winning, overall, someone always wins. Another example is an immortal monkey hitting keys at random on a keyboard: Given enough time, perhaps a googolplex years in the future, the monkey will almost surely have typed the complete work of William Shakespeare at some point.
Let’s look at Pro Tour seasons from this perspective. For simplicity, assume that every season has 3 Pro Tours and that there are 24 players (the current size of the World Championship field) who can bat at LSV’s average. Then, the probability that at least one of those 24 would Top 8 all 3 Pro Tours in the season is 1-(1-0.0015)^24=0.035 or 3.5%. Given that, the probability of seeing 18 consecutive seasons without such a triple-Top 8 is 1-(1-0.035)^18=0.47 or 47%, which means that the probability of seeing at least one triple Top 8 in that 18-year span is 53%. And that’s actually a huge underestimation because it doesn’t take into account additional players, streaks spanning multiple seasons, and extra Pro Tours.
In other words, even though it was an impressive feat for Luis, it was very likely that someone would do it by now. It’s just awesome that it happened to one of the best ambassadors of the game.