What's the theoretical maximum score on the TV show Jeopardy? To get the maximum possible score, one has to ring first on all answers (or have a different first ringer answer incorrectly), answer everything correctly, bet the maximum amount on all occasions, and then a luck factor plays in. The Daily Double clues are located at the optimal spot. If all of the non-luck factors are obtained, what's the probability that yields all of the lucks in place to obtain the theoretical maximum score?
In the first round, there are 6 categories, with each with $200, $400, $600, $800, and $1000 clues. Add them up and multiply it by six: 6*(600*5) = 18,000. However, there is one Daily Double clue in this first round. Since the Daily Double clue annuls the monetary value originally associated with the clue, the Daily Double has to be hidden behind a $200 clue to minimize the annulling, and thus maximizing the overall score. Subtract 200 from 18,000 leaves 17,800. If everything is wagered on Daily Double and the answer is correct, that leaves 17,800*2 = $35,600 just from the first round.
In the Double Jeopardy round, everything is doubled: 6 categories, each with $400, $800, $1200, $1600, $2000. Furthermore, there are two Daily Doubles. Start with the $18,000 figure calculated from the previous round. Double that, since it's Double Jeopardy: 18,000*2 = 36,000. Now, we need to backtrack two of the $400 values to account for the Daily Double. That leaves 36,000 - 2*400 = $35,200 gained purely from Double Jeopardy, without the Daily Doubles. Don't forget to add the amount from first round, and that leaves 35,600 + 35,200 = $70,800. Now for the two Daily Doubles, as well as the Final Jeopardy, everything is wagered and answer correctly. That is 3 times of doubling the score: 70,800*2^3 = 70,800*8 = $566,400.
Now the question is, if suppose someone can ring first on all answers, answer everything correctly, and still bet the maximum amount on all
occasions, what's the probability that the Daily Doubles will be located in the correct spot to allow this theoretical maximum score? Well, in the first round, it has to be hidden in one of the $200 clues. That's a 1/5 chance of that happening. Independently from that, the chance of the two Daily Doubles both hidden under $400 in Double Jeopardy is (6/30)*(5/29). The first one can be in any of the six $400 slots, out of the 30 overall clues. After that's taken care of, the second one has to be in any of the five remaining $400 slots, out of the 29 overall remaining clues. Therefore, the overall probability is (1/5)*(6/30)*(5/29) = 0.00689655172, which is exactly 1/145.
So overall, if someone can ring first on all answers, answer everything correctly, still bet the maximum amount on all
occasions, and has the 1/145 luck chance, it is possible to get the theoretically maximum score of $566,400 on Jeopardy. To give a comparison, that value is over 1/5 of the amount Ken Jennings earned during his record-setting 74-win streak.
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