Ask The Wizard #314

Let me set the stage. On the June 3, 2019 James was within easy reach of breaking the record for total money won in regular games, which still stands at $2,520,700. James average win per game was much more than what he needed to break the record. So all eyes were watching on June 3 to see the record broken.

Instead, what happens is not only does James not break the record, but he loses. The winner, Emma, played a very strong strategic game as well as being good with the buzzer and simply answering correctly. She played just as James normally did. Going into Final Jeopardy the scores were:

• Emma — $26,600
• James — $23,400
• Jay — $11,000

In these situations, where second place has more than half of first place, and third place does not, it typically comes down to first and second place choosing to go high or low with their final wager. A high wager for first place is enough to lock in a win if correct. To be specific, two times the second place score less the first place score plus one dollar. That is exactly what Emma did with a wager of 2×$23,400 - $26,600 + $1 = $20,201. Most of the time, the first place player does this.

However, James didn't know what Emma would do when deciding his wager. The following table shows who would win according to what combination of wagers.

Click on image for larger version.

If Emma wagers at least $20,201 then she locks in a win if correct.

If Emma wagers low then she will win if either (a) James wagers low or (b) James wagers high and is wrong.

If James wagers high then he wins if (a) Emma goes high, Emma is wrong, and James is right, or (b) Emma goes low and James is right.

If James wagers low then he wins if Emma goes high and is wrong.

If perfect logicians were playing, both would randomize their decisions. However, rarely does the leader go low in these situations where he/she can be caught. If James anticipates Emma to go high, then he absolutely should go low. This way he doesn't have to get Final Jeopardy right to win, he just has to hope Emma blows it.

James actual bid was the correct amount to cover Jay if Jay bet everything and was right: $23,400 - 2×$11,000 - $1 = $1,399, which satisfied as a low wager for purposes of beating Emma.

If correct, James would get an extra $1,000 for coming in second, compared to third.

In conclusion, I completely reject the conspiracy theory that James threw the game. He played the right way and lost due to a combination of playing a strong competitor and what most people would call "bad luck."

External Links

• Jeopardy hall of fame
• James Holzhauer on Jeopardy — Discussion in my forum at Wizard of Vegas.

It is easy to choose these two, as probably the two most famous:

• 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... = π/4
• 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... = π^2/6

Based on that advantage and hands played, the expected win is 2015 units. Assuming a standard deviation of 1.1 per hand, a standard deviation on the entire play is 1,561. The difference between your actual win and expected win is 3,490. That is 3,490/1,561 = 2.24 standard deviations. The probability of results that bad or worse is 1.27%.

That is true, with 23 random people, the probability of at least one pair of people have a common birthday is 50.73%. This ignores leap day and assumes everybody has an equal chance of being born on each of the other 365 days (which is not actually the case, spring and fall birthdays are slightly more common).

The tables in answer to your question are quote long, so I'll put them in spoiler tags. Click on the buttons for the answers.