The Jeopardy 3-way tie

Johnny McEldoo · #1 03-17-2007, 01:47 PM

Wondering if some of you probability wizards could help me with this...

I didn't see the episode but today I heard on the radio that at the end of Final Jeopardy there was the first 3-way tie between contestants in the history of the show. The person doing the report said that someone came up with the probability that such thing could happen was 1 in 25 million.

This seems to be an interesting question because since the dollar amounts are relatively consistent through the first two rounds, a tie of this nature is somewhat feasable. However, when you get to final jeopardy, players are allowed to bet any amount up to the amount they have so there could be a wide range of possible outcomes. Also, in most cases the players are usually attempting to have a final amount that is different (higher) than their opponents so that the chance that all come out with the same end result less likely. With the only exception being that they all ended up with 0 because they all bet their entire amount and got the question wrong...

Adding in the Final Jeopardy to the equation, it seems extreemly difficult to come out with a probability because of the huge amounts of possible outcomes as well as having to figure out how all 3 players are thinking about the result.

Anyone able to help me understand how this guy could come up with a number of 1 in 25 million?

I'm not concerned about the actual number he came up with but the process that one would use to approach this type of problem.

Sometime I hate when I get things stuck in my head [img]/images/graemlins/smile.gif[/img]

PairTheBoard · #2 03-17-2007, 11:01 PM

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Anyone able to help me understand how this guy could come up with a number of 1 in 25 million?

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Maybe he figured the Final Jeopardy dollar amounts tend to be about equally distributed within a $5,000 range. That would produce the estimate that the chance the second and third player match the first player's number is

(1/5000)(1/5000) = 1 in 25 million.

If you assumed the Final Jeapardy dollar amount was something more like normally distributed I'm not sure how you would compute it. I imagine the pros here would know. Trying to factor in strategy decisions for the final bet would probably be pretty tough. I suspect the figure they came up with was something quick and dirty like what I gave above.

PairTheBoard

David Steele · #3 03-18-2007, 01:03 AM

I doubt that 1 in 25 million is anywhere near correct.

Surely the guy in the lead deliberately bet the amount that would give him 16000 and a tie if the others bet everything (8000, which they both did). It's a good idea to get the same people back that he can beat rather than random, possibly good players.

The requirement then starts with just getting 2 players tied which is relatively common.
Also consider points are almost always multiples of 200 ( exception is rare daily double bets), so its more like the scores are 80, 80 , 122 ( or whatever the lead was).

D.

Orlando Salazar · #4 03-18-2007, 05:50 PM

unless when people tie, everyone gets the cash, the leader before the final question is a total IDIOT!!!!

David Steele · #5 03-18-2007, 08:24 PM

They all 3 get the full amount, plus he gets to play the weak players again, so no idiot.

D.

SumZero · #6 03-19-2007, 05:12 AM

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They all 3 get the full amount, plus he gets to play the weak players again, so no idiot.

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Yeah, I always wondered why the leaders normally bet to win by $1 rather than play for the tie.

TomCollins · #7 03-20-2007, 01:26 PM

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They all 3 get the full amount, plus he gets to play the weak players again, so no idiot.

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Yeah, I always wondered why the leaders normally bet to win by $1 rather than play for the tie.

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If your opposition is strong, you don't want to play them again.

Johnny McEldoo · #8 03-20-2007, 04:55 PM

does anyone know if they got to keep their winnings when they tied? i thought i heard they all just got to come back the next time for a rematch but i could be wrong.

if that's the case it's not such a big of a blunder by the chip leader as i thought...

alThor · #9 03-20-2007, 10:36 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
They all 3 get the full amount, plus he gets to play the weak players again, so no idiot.

[/ QUOTE ]

Yeah, I always wondered why the leaders normally bet to win by $1 rather than play for the tie.

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If your opposition is strong, you don't want to play them again.

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And if you're in the lead, your opposition isn't strong.