|Dock of the Bayes
February 2005: I actually learned some things about mathematics through these columns. Odds are that I was the only one.
WCAC/QuAAC Corner: (Sitting on) the Dock of the Bayes1
By David Fleming, executive director of faculty development and assessment
Gary Franchy, division chair, math
DF: Gary, I've had well over a month now to think about our promised QuAAC topic for this month: conditional probability. I'm ready to go it alone.
GF: This will all end badly.
DF: Oh, ye of little faith. Because I'm such a sports fanatic, I've been able to keep track of many examples of how conditional probability can lead to the misinterpretation of statistics.
GF: This could be good. I'm going to let you keep going in the spirit of "give him enough rope."
DF: No, I won't use any Boxing examples. I don't get that sport. But, take the Super Bowl...
GF: I don't think you can use "Super Bowl.
DF: You didn't even let me finish.
GF: No, I think you have to say "NFL championship game"; "Super Bowl" is trademarked and I don't want to get sued. Do you have the "expressed written consent of the National Football League"? I don't think so!
DF: I knew the stress was getting to you.
GF: Hey, we live in a world where Barbra Streisand sues an environmental group.
DF: Fine, as the "NFL championship game" goes in to the 4th quarter, the announcers almost always say, "historically, the team who is leading going into the 4th quarter has won X percent of the time," "x" being, of course, some astronomically high percentage.
GF: I see what you're thinking. And since most Super Bowls are blow outs . . .
DF: Exactly, the conditions of the game change the probability of that 4th quarter lead turning into a win dramatically.
GF: Well, like Scott Norwood, you were a little right on that one.
DF: Or, one of my favorite stats from baseball, the incredibly high ratio of runs-batted-in to at-bats for a player in bases loaded situations.
GF: O.k., I'll keep playing.
DF: The guy makes contact and he will probably get a run-batted-in, and any hit will probably mean two or more runs-batted in. The conditions are favorable for the chance of an RBI.
GF: Well, you're not completely in left field on that one.
DF: Wait, I'm just getting started. Let me start talking about permutations.
GF: Oh boy.
DF: Wait, let me start talking about permutations. I'm just getting started.
GF: Stop that!
DF: Wait, just let me start talking. I'm getting started about permutations.
GF: Enough! It's time I put my foot down.
DF: Be careful where you put it.
GF: Good point. Yours always ends up in your mouth.
DF: Fine. Take over, then.
GF: O.K. but let me just say that you did bring up a couple valuable learning points.
GF: But, your stuff had almost nothing to do with conditional probability.
GF: What you focused on was the conditions in which information is compiled.
GF: That's known as "context" and it is necessary to understand the circumstances so we can make the best use of the data we collect.
DF: At least it wasn't a total waste; what was my other point?
GF: Some things are best left to the professionals.
DF: Ouch. What should I have said?
GF: Well, the basic idea of conditional probability is that additional knowledge can help us refine initial probabilities.
DF: Didn't I say that?
GF: Not really. In your football example you should have compared the fourth quarter probability to the chance of that team winning before the game started.
DF: How about my baseball example?
GF: I'd rather start from scratch. Baseball is notorious for tracking performances in every situation, like differences between playing on grass and playing on artificial turf?
DF: And night games versus day games, against left-handed pitchers and right-handed pitchers, etcetera.
GF: Exactly! A "300" hitter isn't always a "300" hitter. A manager uses all of this information to position the infield or outfield, how to pitch to a batter, whether to change pitchers to face a batter...
DF: But won't the other manager know the same things?
DF: Well, if we were the managers for opposing teams, wouldn't I be able to tell what you are going to do?
GF: Well, if you are looking at my players as they shift in the field, I'd think so.
DF: Very funny, but I'm talking about things like pitch selection. There was that scene in Bull Durham where Kevin Costner tells the batter what pitch is coming next and the batter hits a home run.
GF: More conditional probability! The chance that a batter will get a hit does increase if they are expecting the type of pitch thrown.
DF: Yes, but back to my point, if I know what you are going to do...
GF: And I know that you know that we know...
GF: It's risk management; managers try to avoid situations that leave them the most vulnerable. As far as pitch selection goes, they will still throw some fastballs to a batter that is a good fastball hitter, to prevent them from waiting on a particular pitch.
DF: And in football, a team may defend more against the run if the opposing offense doesn't have a strong passing game.
GF: I didn't know you were a Lions fan.
DF: Don't go there. Do you have any more examples for us?
GF: Sure, have you seen a poker tournament on TV lately?
DF: I love it when you get all topical.
GF: While watching a Texas Hold'em game, the network displays probabilities next to each person's cards. As more cards are revealed...
DF: The Flop, the Turn and the River.
GF: Very good, Skippy. As more cards are revealed, the probabilities change.
DF: Speaking of card playing, is that the reason that casinos use multiple decks at their Black Jack tables?
GF: Yes, the more decks that are in play the more difficult it is for a player to "count" cards and recompute probabilities in a timely enough fashion to give them an edge.
DF: Even for someone with a strong math background?
GF: All I can say is that I'd never try it.
DF: Why not?
GF: Because I like to lose my money fair and square.
1 — Ask your local mathematician/statistician if you are truly curious about the title