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How Much Money Can You Win in a Single Episode of Jeopardy!?

...and why it's unlikely anyone will actually win it.

Jeopardy

Jeopardy! host Alex Trebek appeared on The Nerdist Podcast recently. It’s a really interesting interview, and if you have an hour and a half to kill, it’s a great way to do that. At one point host Chris Hardwick wondered what the theoretical maximum a player could win in one episode was, so we did the math.

According to both Wikipedia and Trebek on the Nerdist episode in question, the current record for most money earned in a single game of Jeopardy! is $77,000. That’s impressive, but if a contestant is a combination of incredibly lucky, quick-witted, and quick with the buzzer they could potentially make a lot more.

First, a quick primer if you’re unfamiliar with Jeopardy! or if you haven’t watched in a while. I’ll block quote it so it’s easy to skip ahead if you’re already familiar.

There are three rounds; Jeopardy, Double Jeopardy, and Final Jeopardy. Each of the first two rounds has six categories of five clues each, and Final Jeopardy is a single clue. Each category in the Jeopardy round has questions with ascending dollar values of $200, $400, $600, $800, and $1000. Those double in Double Jeopardy (hence the name) to $400, $800, $1200, $1600, and $2000.

In the Jeopardy round one clue is secretly a “Daily Double” that allows the player who selects it to wager any amount of their current total on getting the answer right but does not award the value of the clue on the board. A “True Daily Double” is when a player wagers their entire sum on the question. If they get it right, their total is doubled. There are two “Daily Double” clues in the Double Jeopardy round. Players do not know which clues are designated as the Daily Doubles until the clue is selected in play.

In the Final Jeopardy round players wager any amount of their total winnings after hearing only the category of the clue, and are again able to double their winnings if they get the right answer.

Getting the theoretical maximum is more involved than simply adding up the dollar values for all the questions because of the wagering allowed by the Daily Doubles and the Final Jeopardy round. A player lucky enough to hit all three Daily Doubles could in theory double their total four times in the game with Final Jeopardy being the fourth chance.

Another factor is that all 30 questions in each of the first two rounds would have to be answered, which doesn’t always happen because of the game’s clock. Rounds frequently end with clues still up on the board, but that’s not always the case. We’ll assume our theoretical super-winner gets through all 60 questions and gets every answer correct.

As for the dollar value itself, in the Jeopardy round the value of all 30 clues equals $18,000, but remember we need to eliminate the value from one question for the Daily Double. We’re eliminating a $200 clue because it takes away the least value from the possible money earned. If our lucky super contestant gets all 29 of the other non-Daily Double questions correct they will have $17,800 which they can double on the last (Daily Double) clue of the round for $35,600.

The Double Jeopardy round clues total $36,000 (double that of Jeopardy) but we need to eliminate two of the $400 for the Daily Doubles. That means our super contestant would have $70,800 after answering the 28 non-Daily Double questions of the round. ($35,200 from the Double Jeopardy round plus the $35,600 that carried over from the Jeopardy round.)

If they went for two True Daily Doubles in a row with that $70,800, they could quadruple it to $283,200 going into the Final Jeopardy round where they again have a chance to double their winnings.

So, in terms Jeopardy! fans will understand:

This is the total amount of money it is mathematically possible to win in a single game of Jeopardy!

What is $566,400 Alex?

If it’s possible to clear half a million dollars in one shot, why is the record only $77,000?

It’s because of the game’s Daily Doubles. To hit the absolute maximum, the player doesn’t just need to hit all three Daily Doubles, they need to hit them last and on the lowest-value clues on the board. That’s $200 in the Jeopardy round and $400 in Double Jeopardy. With 30 clues per round, and six categories on the board, odds of all three Daily Doubles being in the lowest value spots and being selected last aren’t great.

In the Jeopardy round there’s a 1 in 5 (6 out of 30 really) chance that the Daily Double is one of the lowest-value questions, and a 1 in 30 chance of any single clue being the last on the board, so the odds of the last clue on the board being both the lowest value AND the Daily Double in the Jeopardy round is 1 in 150.

The odds for Double Jeopardy are worse, because you need the two Daily Doubles to BOTH be the lowest value AND the last two picked. You have just a 1 in 12,615 chance of that being the case in the second round. Combine that with the requirement that the Jeopardy round also end on the Daily Double and the odds are 1 in 1,892,250. That means that it’s only possible to win the absolute maximum amount of money 1 out of every 1,892,250 games of Jeopardy! played.

According to the Trebek interview, Jeopardy! runs five new shows per week 46 weeks out of the year for a total of 230 shows each year. That means at the current shooting schedule of the show this chance comes up once every 8,227 years.

(via Nerdist Podcast with special thanks to Brian Tickle for being my math consultant, image via Jeopardy!)

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