As a former math teacher and academic teach coach from several years ago, and also as a bit of a trivia fan, two of my areas of interest are math and the TV game show Jeopardy.  In a previous blog post from September, I wrote about Jeopardy Math.  In that post, I explored the mathematical question of what the highest possible score would be in a game of Jeopardy.  Here, as I write this one, I have recently watched the conclusion of Jeopardy’s 2021 Tournament of Champions.  A couple of events from this year’s tournament have prompted me to share some more Jeopardy thoughts, much of which are math-related.

Before I get too far into this, I should mention that this post contains spoilers.  So, if you are a fan of Jeopardy and have not watched this year’s Tournament of Champions, but you intend to, and you don’t want to know what happens until you actually watch it, you may not want to read any further just yet.  Watch it first, then come back and read this.  (I believe all the episodes can be found on YouTube.)  Otherwise, if you’re okay with finding out information before (or without) watching the tournament, you may go ahead and read further.

I do have a couple more disclaimers I would like to make before I proceed, which I also made in my original Jeopardy Math post.  First, I realize that the clues are technically the “answers” and the responses are technically the “questions.”  But I may refer to clues as questions and responses as answers, because that’s how everyone thinks of it anyway.  So don’t bother pointing that out.  Second, I also realize that Jeopardy has an exclamation point at the end, but I don’t include the exclamation point.  So don’t bother pointing that out.

Now that I have all that out of the way, I wanted to spend some time on the math involved in Jeopardy, using the recent Tournament of Champions as an example.  I will also cover some thoughts on Alex Trebek and the Jeopardy hosting situation at the end.  But before I do, the main point which drove me to write this is, I would like to see Jeopardy contestants do a better job of wagering.  These contestants know so much more than I do and would completely obliterate me if I ever appeared on the show, yet I believe I would make a good “Daily Double Wager Consultant”, if such a position existed.  Perhaps Jeopardy can create a rule where a contestant gets to call a timeout when confronted with a Daily Double and talk to a coach about how much to wager.  I would like to be that coach.  I assume I would get 10% of their winnings.

Specifically, there were two Daily Doubles this week in the Tournament of Champions where I believe the contestant could have made a better wager, and actually cost themselves the game by not doing so.  I am not writing this to embarrass anybody or be overly critical, so I will stay away from naming any names (as if any of these contestants or anyone else from the show are reading this anyway).  However, I would like to reference these two specific examples from this tournament to illustrate my point.

Example #1

The first example occurred on the third semifinal game that aired on Wednesday, May 26.  In that episode, one of the contestants had a slim lead late in the game when he came across the final Daily Double and was faced with the critical decision of how much to wager.  Specifically, he was ahead $16,800 to $14,200 at the time, for a lead of $2,600.  (The third contestant had $2,000.)  There were also nine clues after the Daily Double still showing on the board: all six $400 clues and three $800 clues.  Now, how much should our leader wager?

Wagering on Daily Doubles often seems to be a gut feeling.  However, given the specific situation here and the late stages of the game, I believe there is an optimal wager that can be quickly calculated.  The way I see it, there are two main goals when playing Jeopardy.  Ideally, the first goal is to have more than double the score of your closest competitor going into Final Jeopardy, because this guarantees you the win.  But, if this can’t be done, or is too risky to try (as was the case here), then the next goal is to do what you can to at least have the lead going into Final Jeopardy.  Having the lead means almost everything, as it guarantees you the win if you answer Final Jeopardy correctly, with the appropriate wager, of course.  So, in this situation, we want to figure out how much to wager on the Daily Double to where a correct answer clinches the lead going into Final Jeopardy.

Remember, our leader has a $2,600 lead at this point.  There are nine clues on the board, including all six $400 clues (totaling $2,400), and three $800 clues (also totaling $2,400).  So, there is $4,800 left on the board.  Putting ourselves in the leader’s shoes, if we can just take a $4,801 lead on this question, we clinch the lead at the end of the round, even if the person in 2^nd place answers all the remaining questions.  All we have to do is put the buzzer down for the rest of the round, get the Final Jeopardy question correct, and advance to the finals.  Of course, we also have to consider, would such a wager be worth the risk?  Well, we are ahead by $2,600 right now, so all we have to wager is $2,201.  With a wager of $2,201 and a correct response, we clinch the lead going into Final Jeopardy.  But even with an incorrect response, we are still ahead by $399, and although now faced with a nail-biting finish, we are still in good shape.  So, that would have been my wager: $2,201.

As I was watching this unfold on TV, I was furiously trying to do this math in my head, and I think I could have figured it out and made my wager by stalling for a couple more seconds.  But before I had this answer completely figured out in my head, and at least hoping he wouldn’t wager more than the $2,600 he was leading by, the contestant went ahead and made his wager: $5,000.  I yelled at the TV, “No! Too much!”  Unfortunately, as I have found from my experience in watching sports and game shows, yelling at the TV somehow does nothing to change the events we are watching.  The $5,000 wager had been made.  Although it would have worked out just fine anyway with a correct answer, unfortunately for him, he gave an incorrect response and suddenly found himself down by $2,400.  When Double Jeopardy ended, he was still behind by $1,600.  Both of the top two contestants got the Final Jeopardy answer correct, and the new leader made sure to wager enough to win the game and advance to the finals.

This got me thinking, what if our original leader had only wagered the $2,201 that I would have advised if I had been allowed to call a timeout and consult with him?  Again, the incorrect answer would have still kept him in the lead, although by only $399.  If everything else had played out the same way, he would have still been in the lead by $1,199 going into Final Jeopardy.  And, given that both of the top two contestants got Final Jeopardy correct, I assume he would have wagered enough to win in that scenario.  So, there’s one example where a different wager would have probably resulted in a different outcome.

Example #2

My second example comes from the tournament’s final show which aired on Friday, May 28.  In the Tournament of Champions, the final three contestants compete in a two-day final, and the scores from both games are added together to determine the overall champion.  After the first game which aired on Thursday, the contestants’ scores were: $38,000, $14,500, and $6,400.  Given these scores, the 2^nd and 3^rd place contestants obviously had a lot of catching up to do on Friday’s show to have any hope of winning.  The top two contestants were in the midst of a close game on Friday, when Thursday’s 2^nd place finisher went on a bit of a run early in Double Jeopardy and found herself with a Daily Double.  At the time, she was ahead $10,400 to $6,600.  Under normal game play, with a $3,800 lead, a conservative wager might have been appropriate, but given that this was a two-day event and given her $23,500 deficit from Thursday, this situation called for going “all in”, as former contestant James Holzhauer would say.  A risky move, to be sure, but one that was made necessary by the deficit she had to make up.  Her wager: $2,200.  She got it correct, but could have earned $8,200 more if she had wagered all $10,400 she had at that time.

Going into Friday’s Final Jeopardy round, Thursday’s leader had $20,400 on Friday, while Thursday’s 2^nd place finisher had $19,000 on Friday.  So, during the commercial break between Double Jeopardy and Final Jeopardy, I did some calculations to figure out two things.  First, did the leader have to wager anything in Final Jeopardy, or had he already clinched the championship?  And second, what would the scenario have been if the 2^nd place contestant had wagered everything on the Daily Double (an extra $8,200) like I was hoping she would?

For the first question, we look at the maximum possible score the 2^nd place contestant can achieve.  She had $19,000 so far on Friday, and if she were to double that, her maximum Friday score would be $38,000.  Add that to her $14,500 from Thursday, and we get a maximum possible score of $52,500.  The leader had already scored $38,000 on Thursday and had $20,400 at that point on Friday, for a total score of $58,400.  So, he had indeed clinched the championship, as long as he wagered nothing (or very little) in Final Jeopardy.

But what if the 2^nd place contestant had wagered an additional $8,200 on that Daily Double earlier?  While some may point out that we don’t technically know if everything that happened after that Daily Double would have happened exactly the same way, I also don’t see any reason to doubt that it would have.  The only variable might have been the final Daily Double, but Thursday’s winner got it, wagered everything ($7,800), and answered correctly.  I imagine this is still what would have happened in our alternate scenario anyway.  So, I’m going to say everything else would have been the same and go with it.  So, if everything else had turned out the same, except for the 2^nd place contestant having an extra $8,200 to work with, here’s what would have happened:

The 2^nd place contestant would have been in the lead on Friday with $27,200 going into Final Jeopardy.  Her maximum possible Friday score would have therefore been $54,400.  Added to her Thursday score of $14,500, her maximum possible total would have been $68,900.  We already figured out earlier that the leader was sitting on a total score of $58,400.  Instead of having already clinched the championship, this would have then forced the leader to wager something in Final Jeopardy and get the answer correct to win.

As it turned out, the 2^nd place contestant got the Final Jeopardy answer correct, and the 1^st place contestant got it wrong.  Given the actual situation, this didn’t matter because the championship had already been clinched (since the leader was smart enough to bet a paltry $12).  But in the all-in scenario above, this would have made a difference and turned the tables, and we would have had a different champion.  This made for the second instance during championship week where I could have helped a contestant win simply by advising of how much to wager on a Daily Double.

Sure, it’s easy to simply look back and say, well, this person got the Daily Double right; therefore, they should have wagered everything.  Or, this person got the Daily Double wrong; because of this, they should have wagered as little as possible.  Just like in football when a team decides to go for it on 4^th down and the announcer asks, “Do you like this decision?” and the commentator says, “I’ll tell you after the play,” waiting to see the result of the play before either praising or criticizing the decision.  However, this is not a results-based analysis.  I’m not using the fact that a contestant got an answer wrong to say they should have wagered less, or using the fact that a contestant got an answer right to say they should have wagered more.  Rather, I’m exploring the logic of the wager itself.  This is about maximizing your probability of winning.  If you are up late in the game and can calculate a wager to put the lead out of reach without risking giving up the lead, do it.  On the other hand, if you’re down by $19,000 in a championship match and have to go for broke to have any chance at all, do it.  Again, I don’t want to be overly critical; these contestants are very bright and are making spur of the moment wagers in a high-pressure situation.  But from where I sit, some Jeopardy wagers can be improved.  It’s about taking necessary risks while avoiding unnecessary ones.

Example #3

Speaking of unnecessary risks, these are not the only two examples I have seen of suboptimal wagering on Jeopardy.  They just happened to be two instances that I saw during the Tournament of Champions this past week.  I have seen way more glaring examples on previous occasions.  A few weeks ago, I happened to see a Saturday rerun that originally aired in probably 2019 or 2020, with Alex Trebek.  One contestant had the lead going into the last clue of Double Jeopardy, and was presented with the final Daily Double.  I don’t remember how much she was ahead by, but seeing as how it was the very last clue before Final Jeopardy, common sense should dictate to wager an amount low enough to not risk giving up the lead.  (The alternative would be to wager enough to try to more than double your opponents’ score, which is super-risky, but I wouldn’t criticize it.)  This contestant, however, needlessly wagered an amount which accomplished nothing except putting her lead in jeopardy (pun intended).  I yelled “No!” at the TV, but again, that never seems to change anything.  It didn’t matter because she got the answer correct anyway, kept the lead, and went on to win.  However, it served as another example of a situation that makes me smack my forehead and say, “Whyyyy?”

Example #4

Another wagering faux pas I would like to stop is when 3^rd place contestants actually have a shot to win going into Final Jeopardy because of how close the top two competitors are, but then needlessly wager everything and end up with $0.  For example, in an episode earlier this year hosted by Ken Jennings, the top two contestants were tied at $18,800 going into Final Jeopardy.  This created a scenario where both contestants were likely to wager everything out of necessity, thus creating the very real possibility that both would get it wrong and end up with $0.  The 3^rd place contestant had $10,000, so simply sitting on the $10,000 and wagering nothing would have been the way to go, in case both of her competitors wagered everything and lost.  (An argument can also be made for wagering $8,801 in this situation, although I won’t go into painstaking detail as to why.)  Essentially, any wager other than $10,000 would have been acceptable.  However, this contestant wagered the full $10,000, got the answer wrong, and wound up with $0.  This would have been a real shame if the other two players had also ended up with $0.  It ended up not mattering, as both of the other players got their answers right and had to go to a tie-breaker, making the 3^rd place wager irrelevant, but it’s the logic, not the result, that I’m concerned with.

There are other examples of suboptimal wagering on Jeopardy that I could research and discuss on here, but I think I’ve made my point.  I should probably write a book on Jeopardy wagering sometime though.  Then I can sell it and make some money that way.  Money that I’m losing out on by not being a Daily Double Wagering Consultant.

“Math Champs”

Meanwhile, I would also like to mention that math is relevant not only to Jeopardy wagering, but also to answering questions on Jeopardy as well.  On Friday’s Double Jeopardy round, one of the categories was “Math Champs.”  So, in the final stages of the tournament with the championship on the line, we were treated to five questions about math (one of which none of contestants knew, but I did).  The eventual winner essentially clinched the championship with his $7,800 wager and correct response on the Daily Double clue on Math Champs: “Also known as Leonardo of Pisa, he knew a lot of numbers; not just 1, 1, 2, 3, 5, 8…”  So, math teachers, the next time one of your students asks you, “Why do we have to know this?”, here’s one more answer to add to your arsenal: “Because you might be on Jeopardy and have an entire category about math.”

Jeopardy Host

One final thought on the Tournament of Champions: I thought Buzzy Cohen did a great job as guest host for the tournament.  He was very articulate and seemed to be in control, as if he had years of prior experience hosting the show, even though he had none.

For the record, I would like to see Jeopardy give the full-time hosting duties to Ken Jennings.  I thought he stepped in very well during those first few weeks doing the impossible job of replacing Alex.  But if they’re not going to give it to Ken, then I’d like to see them give the job to Buzzy.  I thought he was that good, and I think either of the former contestants bring a certain credibility to the show.

We haven’t seen all the guest hosts yet, so who knows what will happen.  In fact, I haven’t even seen all of them that have been on.  Are we just trotting out everyone now?  Can I guest host?  I have experience as a quizmaster.  Jeopardy, if you’re reading this, I’d like to guest host for a week.

Of course, nobody can really replace Alex, and Jeopardy will really never be the same without him, but the show must go on.  Can we just have an election for full-time host and move on?  Vote for Ken.

Finally, I would be remiss if I didn’t mention Alex.  Obviously, everyone was saddened by his passing a few months ago.  I’ll just put in a little plug here – if you’re looking for a new good book to read, I recommend his recent autobiography that came out last year shortly before he died, “The Answer Is…”  It’s an interesting read.  He recalls his years growing up in Canada and his life before Jeopardy, and writes about a wide variety of subjects, including philanthropy and his battle with pancreatic cancer.  He also shares some insights and tidbits from the show, including memories of certain contestants.  Most notably, he shares his philosophy that the show wasn’t about him; he wasn’t the star.  Rather, the show was about the contestants.  That’s what made him such a beloved host.  So, hopefully whoever the next host is carries that same outlook.

Conclusion

So, there you have it.  Those are my thoughts on Jeopardy wagering, strategy, and math; on the Tournament of Champions; on Alex and the guest hosts.  In conclusion, if you ever make it onto Jeopardy and want someone to coach you on how to wager on Daily Doubles and Final Jeopardy, I’m available for further consultation at 10% of your winnings.  If you’re from Jeopardy and want to bring me on as a guest host for a week or as a Daily Double Wager Consultant or Final Jeopardy Wager Consultant, or if you’d like to discuss my recommendations for the next permanent host of the show in more detail, feel free to contact me and we can talk.  For anyone else out there, if you’re looking for something good to watch on TV, may I recommend all the episodes from the recent Jeopardy Tournament of Champions.  (Again, I’m sure you can find these on YouTube.)  If you’re looking for a new book to read, I recommend “The Answer Is…” by Alex Trebek.  If you’re looking for another new book to read after that, just hang on until I get my unfinished (un-started) book on Jeopardy wagering published.  Or you can just turn all my blog posts into a book and read that.

“I’m curious about everything.  Even subjects that don’t interest me.”  –Alex Trebek