My first job was at UCLA in 1976. The legendary Ted Harris who wrote one of the first books on branching processes, found a tractable subset of Markov chains on general state space that bears his name, and invented the contact process was there at University of Southern California. USC was only about 20 miles away and Ted Cox was there from 1977-1979, so I would often go over there on Friday afternoon for the probability seminar. On weeks when there was an outside speaker, Ted and his wife Connie would have little after-dinner desert parties in their house at the southeastern edge of Beverly Hills. One of Connie’s favorite things to say was, you guessed it, “There is no such thing as probability.” To support this claim she would tell stories in which something seemingly impossible happened. For example, one evening after eating at a restaurant, she realized while walking to the exit that she had left her purse at the table. She went back to the table to retrieve it and along the way saw an old friend that she had not seen in many years. She would never had seen the friend unless she had forgotten her purse. The punch line of the story was “What is the probability of that?”
The connection with basketball is that this year’s March Madness seems to violate some of the usual assumptions of probability.
The probability of events such as black coming up on a roulette wheel do not change in time. Of course, the probability a team wins a game depends on their opponent, but we don’t expect that the characteristics of the team will change over time. This is false for this year’s Duke Blue Devils. They lost to UNC in coach K’s retirement game on March 5, and sleepwalked their way through the ACC tournament needs late game surges to beat Syracuse and Miami, before losing to Virginia Tech in the finals.
They won their first game in the NCAA against Cal State Fullerton. This was a boring game in which the difference in scores being like the winnings of a player betting on black every time. Playing against Michigan State, it looked like it was all over when Duke was down by 5 with a minute to play but they rallied to win. In the next game against Texas Tech in a late game time out, the players convinced coach K to let them switch from zone defense to man-to-man. If I have the story right, at that moment coach K slapped the floor, and then the five players all did so simultaneously, an event of intense cosmic significance, and Texas Tech was done for. Maybe the French theory of grossissement de filtration can take account of this, but I am not an expert on that.
You have to take account of large deviations. In the first round #15 seed St. Peter’s stunned the nation with an upset victory over #2 Kentucky, and then beat #7 Murray State to reach the Sweet 16 where they played Purdue. St. Peters plays with four guards and early in the game substituted five new players for the starting five. The four guards buzzed around the court annoyed the players that were bringing the ball up the floor and generally disrupted Purdue’s game. To the 7’4” center they were probably like the buzzing of bees that he could hear but not see since they were so far below him.
Basketball is a great example of Lévy’s 0-1 law: the probability of a win, an event we’ll call W, given the current information about the game (encoded in a sigma field Ft) converges to 0 or 1 as t tends to ꝏ (which is usually 40 minutes but might include over time). Late in the game this quantity can undergo big jumps. Purdue was down by 6 with about a half-minute to play and desperately needed a three point shot. The player with the ball turned to throw it to a player who he thought would be nearby and open, only to find that the player had decided to go somewhere else, and suddnely the probability dropped much closer to 0
Games are not independent. Of course, the probability a team wins a game depends on their opponent, but even if you condition on the current teams, the tournament does not have the Markov property. On Thursday March 24 Arkansas upset #1 seed Gonzaga. After this emotional win and with little time to prepare they played Duke on Saturday and slowly succumbed. In a display of coach K’s brilliance “Duke won the final minute of the first half” increasing their lead from 7 points to 12. Even though the game was a martingale after the end of the first half, the L2 maximal inequality guaranteed than a win was likely.
The high point of the Duke-Kansas game came about two minutes into the second half when an Arkansas three point shot bounced off the rim and ended resting on top of the backboard. A quick thinking (and very pretty) Arkansas cheerleader got up on the shoulder of her male partner for their gymnastic routines. Putting her two little pom-poms in one hand she reached up and tipped the ball back down to the floor.
To end where I began What is the probability of that? To use the frequentist approach we can count the number of games in the regular season and in the tournament in which this event occurred and divide by the number of games. The answer would be similar the Bayesian calculation that the sun will rise tomorrow. Using Google and using the curious biblical assumption that the solar system is 5000 years old gives (as I read on the internet) that the probability that the sun will not rise today is 1/186,215. Even though it may seem to my wife that there are this many games, I have to agree with Connie Harris: in March Madness there is no such thing as probability.