Discrete Events and Narrative Lives (Part 1)
Suppose your favorite ballplayer, Allen Average, comes to bat in the eighth inning. His team trails 5-4, there is a runner on third, and two outs. The defending team only needs one out to end the inning. You really want Allen to get a hit, which would bring in the tying run. Now, Mr. Average is not a great hitter, though he plays excellent defense (which is why the manager puts Allen in the lineup most days). Allen’s batting average is almost exactly the league average. Ordinarily, you wouldn’t feel much confidence in this situation, since as an average hitter Allen fails to get a hit more than 70% of the time.
But today is different, you feel. This is Mr. Average’s fourth plate appearance of the game. In his first three at-bats, Allen got a hit each time, and one of them was a double. Allen is having a really fine day hitting! Allen is “seeing the ball well,” as professional players often say. As he approaches the batter’s box, you’re feeling a fine mixture of dread and hopefulness, the heart of baseball fan experience. On this occasion you’re feeling more hope than dread. Today is Allen’s day! He’s going to get a hit—at the very least, you feel he has a better chance today than on other days.
Your feelings deceive you. Sabermatricians have explored this question thoroughly. (SABR is the Society for American Baseball Research. You never may have heard of it, but it’s real. There are thousands, many thousands, of nerdy baseball fans well trained in statistical analysis. They have spent millions of hours researching questions like this one.) They have asked the question this way: take any player who has had a recent string of success or failure up to time x, what are his chances of success at time x+1? Baseball gives multiplied thousands of examples to check, and the result is clear.
A baseball player’s career average, his average up to time x, gives a much better prediction of his chances of success in any particular at-bat (x+1) than his earlier success or failure today. Given that Allen Average has had good success in his first three at-bats today, what should we say about his final at-bat today? Answer: his prior success today is irrelevant; Mr. Average’s chance of success in his last at-bat is approximately equal to his overall success rate. Recent success or failure doesn’t matter.
One way to summarize this lesson is to say that at-bats are discrete events. Each event, we might say, is causally independent of the others. They are affected by a player’s underlying talent, but not by recent success or failure. Today’s narrative (Allen is having a fine day hitting!) has no discernable effect on the immediate future.
Imagine another illustration from a different sport. A basketball player, Tom Terrific, prepares to shoot a free throw. Mr. Terrific is a good shooter; over his career he has made 85% of his free throws. Tonight, though, Tom is not shooting free throws very well; he has made only 6 of 12 free throw attempts. What are his chances of making this next free throw? Statistics from professional basketball answer clearly: Tom’s chance of success on his next free throw (x+1) is indistinguishable from his career average. His earlier failure tonight doesn’t matter.
When it comes to predicting the future, it seems that narrative doesn’t matter much. And this bothers some people. We want to think that since Allen Average is having a good day he has a better chance of getting a hit on his next at-bat. We fear that since Tom Terrific has missed half of his free throws, he is unlikely to make his next one. But we’re wrong.
Sometimes people turn narrative expectations on their head. Since Allen has three hits already, they say, Allen is “due” to make an out. Tom is too good a shooter to keep missing, they say, so he is almost sure to make the next one. Again: not true. The next at-bat, the next free throw—in general, the next discrete event—is independent of recent events. Narrative just doesn’t matter, at least in the short term.
But people live narratives. Even when we know better, we let the story of our lives direct our thinking. Here’s an example.
Once or twice a month, I play Settlers of Catan with some friends. If you’re familiar with Settlers, you will understand my example easily. If you’ve never played, the point will still be clear.
In a game not long ago, I had ten resource cards in my hand when it came to my turn. I rolled a seven, which meant that I lost half my resource cards. (Game rule: anytime a seven is rolled, players with more than seven resource cards lose half of them.) A significant setback! The next time it came my turn, again I had a large hand, and again I rolled a seven. Agh! More resources lost! My next turn: yet again I lost half my cards due to rolling a seven. Three times in a row!
Now dice rolls are discrete events. There is a 1/6 chance of rolling a seven on any particular roll of two dice. By the laws of probability, a person will roll a seven three times in a row, on average, once in 216 chances. So it’s not impossible. It does happen—rarely. But what about the next dice roll, the x+1 roll? Since dice rolls are discrete, independent events, the fact that I had three sevens in a row said nothing about my odds of getting another seven. The chance is still 1/6 of rolling a seven. As a university professor I teach logic. I know this.
But we live narratives. Losing all those resource cards had torpedoed my chance to win the game. I was frustrated. So: just before my next turn, I traded cards with another player, giving up three for one, to trim my hand to seven cards (thus making me immune to the seven rule). That is: I made a bad trade, a trade that further reduced my already tiny chances to win the game, because I feared rolling yet another seven. Obviously an irrational decision! But I was living a story, a frustrating story. Narrative can induce crazy behavior.
By the way, I rolled another seven.