The Nachoman in Mathemagic Land

I emphasize repeatedly to my students that physics is not magic – the natural world obeys straightforward rules whose effects occasionally boggle the mind. But once a year, I choose to do a “magic trick,” and I let the class propose all kinds of crazy explanations for what turns out to be a simple but counterintuitive bit of mathematics.

Here’s the trick, and you can try it at home: I have each student imagine the results of 100 coin flips, writing down the resultant sequence of heads and tails (something like HTTHHHTHTHHTH and so on). Next, each student grabs a handful of pennies, and actually flips 100 coins, writing down the resultant sequence on the other side of the same paper. All the students hand in their papers: one side of each paper consists of a real, random sequence of coin flip results, the other side consists of fake flips. Though the student has identified for himself[1] which side is which, the sides are indistinguishable to me.

Except that I correctly identify the real coin flip sequence for the vast majority of the papers.

The students at first don’t believe that I can do it; very quickly, as I get the first five or six right, they start searching for the “trick.” Did I watch them carefully when they were writing? Did I pre-mark the paper? Did I have an accomplice, or a hidden video camera? No, no, and no. I just played an old Jedi mind trick.[2]

What’s counterintuitive about sequences of random events is how frequently a “run” occurs. It’s human nature to think that, if the last four flips have been heads, the next one probably will be tails. After all, getting five of the same outcome in a row is a rare event. If you flipped exactly five pennies, then they’d all come up the same way only about 6% of the time.

Thing is, we’re NOT flipping exactly five pennies. We’re flipping 100 pennies. And the more pennies flipped, the more likely it becomes that that we see 5 in a row. Sparing you the details, in 100 flips, it is 96% likely that there is, somewhere, a set of 5-in-a-row; it’s 79% likely that there is a set of 6-in-a-row.[3]

All I do, then, to figure out which is the real set of coin flips, is to look for the longest run. Chances are, the students won’t include a run of more than four of the same outcome in a row; chances are, the real flips will include a run of at least five. It’s amazing how well this works, and how much the trick can impress even a surly class of teenaged boys.

Uh, Nachoman, this is a sports column, not the nerd-of-the-month blog. Where’s the relevance?

I was just getting to that… really…

I have seen it suggested[4] that “streaky” shooting in basketball is nothing but an artifact of the behavior of random numbers. By the same logic I used in the coin flip trick, a basketball player who hits 50% of his 3-pointers should, over the course of a 250-attempt season, hit 6-in-a-row at least once. In fact, during the season there’s about a 10% chance that he hits 11-in-a-row.[5] Call it a hot streak if you want, but he’s still got a 50-50 chance of making the next one.

As for baseball, we can estimate the likelihood of long hitting streaks. Consider a .300 hitter with a long career spanning about 2500 games. With some basic assumptions like four at-bats per game, we can calculate about a 7% chance for this player to accomplish a 40-game hitting streak. Not much of a chance, eh? Well, consider that 48 players have played 2500 games in their careers, and 210 players have bested 2000 games. Our 7% estimate suggests that at least 3 and no more than 14 players own 40-game hitting streaks.

Sure enough, I checked it out – six players (all of whom played more than 2000 games) boast 40-game hitting streaks. Math works, folks.

And okay, I promise less math in my next post.


[1] I teach at a boys’ school – no political or grammatical statement is being made by the use of the masculine pronoun here.

[2] As described in “On the run: Unexpected outcomes of random events” by Mark P. Silverman et al, in the April 1999 issue of The Physics Teacher.

[3] And more than half the time, there will be a set of 7-in-a-row. Wow.

[4] Though, to my chagrin, I didn’t take the suggestion seriously enough at the time to write down the source of the suggestion

[5] …and the same likelihood that the player has a similar streak of misses.