Jeopardy! Math

Alex Trebek, the most pompous of all game-show hosts.


As a Jeopardy! fan, I enjoyed this article from 10+ years ago that presents some interesting scenarios about how one should bet in Final Jeopardy. I wonder if actual players on the show ever think that deeply and mathematically about their final bets. Not that the ideas in the article are super mathematically complicated, but still very interesting. Read on.

(And, the author of the article is named Matt Gaffney!)


Math for Jeopardy! Players
How contestants regularly blow their final bet.
By Matt Gaffney
Posted Thursday, July 20, 2000, at 3:00 AM ET

Why do so many Jeopardy! contestants blow it in the final round? Picture this scene, from the March 21, 2000 show: Going into "Final Jeopardy!" Andrew was in the lead with $8,000, Haley was in second with $5,700, and Dave was back in third with $2,700.

If you're Andrew in this situation, deciding on your bet is simple, assuming for argument's sake that the Final Jeopardy! category is neutral, i.e., one you know neither particularly well nor particularly poorly. Andrew's rational path is to wager the minimum he needs to put himself out of Haley's reach—that is, enough to give him twice her current score, plus $1. That's $3,401 in this case, which is precisely what Andrew wound up betting.

For Haley, betting is more complicated. Before I tell you how she should have bet, consider how she did bet. Like most contestants, she took a deep final-scene-of-Thelma-&-Louise breath, bet $5,600, got the final question wrong, and lost. Andrew got it right, won $11,401, and went back the next day. Dave, if anyone cares, bet the house, got Final Jeopardy! wrong, and wound up with nothing.

Here's what Haley should have bet: $299. Notice that the way she actually bet, the only way she could have won is if she'd gotten Final Jeopardy! right and Andrew had gotten it wrong. Obviously, if Andrew answers correctly, the game's over, no matter what Haley does.
By betting $299, Haley gives herself an extra chance. If Andrew gets it right, he still wins, as before. And, as before, if Haley gets it right and Andrew misses it, Haley wins. Here's the difference: If Haley bets $299 and they both miss Final Jeopardy! Haley wins. Her final total would be $5,401, while Andrew would be down at $4,599.

Why can't Haley bet more than $299? Because she has to guard against Dave, whose maximum score, if he bets everything and gets Final Jeopardy! right, would be $5,400. Note that, with correct wagering, Dave is a non-factor in this Final Jeopardy! equation. Even if he bet it all and got it right, he still wouldn't be able to overcome Haley, even if she answered incorrectly.
All this wouldn't have helped Haley in this case, since Andrew answered Final Jeopardy! correctly. But had he missed it, she would've won.

For the player in second place, this all boils down to betting an amount that still gives you the win if both you and the player in the lead miss Final Jeopardy! A wagering-savvy former Jeopardy! champ has labeled this "The Two-Thirds Rule," because the second-place player needs at least two-thirds of the leading player's score going into Final Jeopardy! to be able to pull this off. (Click here for more on the two-thirds rule.)



If the third player is close enough to worry about, as in the example above, you need to guard as much as possible against him. The following scenario from a recent show is a perfect illustration of this principle. Going into Final Jeopardy!, Melizza was in the lead with $7,500. Second was Miles with $7,300, and third was Judy with $5,800.

Again, the leader's bet is easy to calculate, and Melizza did in fact wager the correct amount: $7,101 (again, that gives Melizza twice Miles' score plus $1 if she gets it right). Miles should bet $4,301, while Judy should bet $2,800.

Why? To answer that, we'll only deal with scenarios in which Melizza gets it wrong — because if she bets correctly and gets the answer right, the game's over no matter what.
Miles' bet of $4,301 puts him out of Judy's striking range should he answer correctly, as he'd finish with $11,601, or two times Judy's score plus a dollar.

Judy's correct bet, $2,800, gives her that extra chance that Haley should have had in the first example. By betting $2,800, she can win if all three players miss the last question. With correct bets and all players missing Final Jeopardy!, Melizza winds up with $399, Miles finishes with $2,999, and we'll see Judy again tomorrow as our returning champion, as she's just finished with $3,000. (Naturally, if Judy gets it right and the others don't, she wins anyway.)

Sound too theoretical? Consider what actually happened: All three players missed the final question. Melizza wound up with $399—which was good enough to win, as her opponents had wagered absurdly. Miles bet the insane sum of $7,000, which left him with $300. Judy bet the house and wound up with nothing. Had Miles bet correctly (or even close to correctly—six grand would still have won it for him), he would have been champ. Or, barring that, Judy could have won it herself.

You may be wondering: If the first person knows that the second-place person is going to have this back door, why doesn't he simply bet just enough to outstrip the second place player's more modest bet? Because there's always the possibility that the second-place person will bet it all. And in practice, the person in first place almost always bets $1 or $100 over what he needs to put himself out of reach. It's this tendency that the second- and third-place players should take advantage of.

There are also a few subtleties we're glossing over — for instance, Haley, Miles, or Judy would get a bigger payday if they bet everything and won. But the safer bet provides the extra chance of simply surviving and coming back the next day. And anyway, the basic premise is clear: If you're in second or third place going into Final Jeopardy!, don't just automatically bet it all. Your better chance may be backing into victory.