opusthepenguin wrote: ↑Fri May 31, 2019 12:52 pm
If anyone wants to do the math, here are the scores when James uncovers DD3, says "Oh, this category seemed hard," and bets "only" $5,000
(...)
So... looks like $9600 left on the board? And if James had lost that $5,000, he'd be down to $26,412 or $7,612 more than Megan's doubled score. So if he'd missed, Megan had a chance to prevent a runaway but not a very good one. She'd have to pick up at least 8 of the remaining 12 clues--more if she doesn't get one of the high-value clues. And if she manages that, she's still got to get FJ right while James gets it wrong. The risk may have been unnecessary on James' part, but it was also very small.
Thanks. I don't think it's helpful to use Megan's current doubled score in calculations, as what matters is her final doubled score - effectively, all she gets of the remaining clues needs to be doubled also. What matters is that James has 31,412, Megan has 9,400, and there's 9,600 left on the board.
If James wagers $5, he has either 31,407 or 31,417. If he gets nothing more (I know...), Megan needs to get 6,400 more to break the lock. That's very hard. Not only does she have to get 2/3 of the remaining values, but she absolutely has to get at least one of the bottom two Kowalski clues (otherwise her goal is mathematically impossible). But it is possible. Also, she has no possibility of getting to a 2/3 game, so the best theoretical case for her is to have to get the FJ while James misses. Now how much does James need to add to his score to ensure a lock? Assuming no rebound opportunities, James needs 2,400. Then he would have 33,807 (or 33,817) and Megan could only get up to 16,600 (= 9,400 + 9,600 - 2,400) if she sweeps everything James doesn't get. (Note that you can only get a multiple of 400 from the remaining clues. While it is true that James would have a lock with 2,200, that's not an available amount.) This means if James gets the 2000 clue and one other clue, he has a lock. (Also, if James gets the 2000 clue, Megan has to get literally every other clue. If he gets the 1600 clue, Megan can leave at most 800 to Rob or a TS.) What would you say the probability of a lock is in this case? I'd say it's well above 99%.
If James makes his usual wager, say $9,812, he has either 21,600 or 41,224. Obviously, it's a lock with a get. With a miss, what does he need to secure a lock? Again assuming no rebound opportunities, the equation is 21,600 + x > 2 * (9,400 + 9,600 - x), or x > 16,400/3. In terms of feasible values, he needs at least 5,600, or over half of the remaining values. That's actually more than he got in the game (5,200), though of course Megan didn't sweep all the remaining clues. In this situation, the probability of a lock is considerably diminished, though remember that it applies only if he misses. (Also note that Megan can get within 2/3 in this scenario. James needs to pick up 2,800 to prevent that.)
What wager ensures a lock with a get? Since Megan can at most get 19,000, he needs to have 38,001, for which he needs to wager 6,589. If he does that and misses, he has 24,823 and needs to pick up 4,400 to ensure a lock and 1,600 to ensure a crush.
With James's actual wager (5,000), he still needed 800 to ensure a lock after the get. That's not exactly 100% sure, but is pretty damn close. With a miss, he would've had 26,412 and would have needed 4,000 to ensure a lock and 1,200 to ensure a crush. While pretty close, I'd say this makes the wager clearly inferior to the (calculable in real time only if you're Watson) $6,589 one.
What is the maximum wager that doesn't risk a crush? One that leaves him with at least 28,501, i.e., $2,911. If he misses, he needs to pick up 3,200 (exactly 1/3 of the remaining values) to achieve a lock. If he gets the DD, he sits at 34,323 and needs 1,600 to ensure a lock (or, alternatively, if he sits pat, Megan needs 8,000 of the remaining 9,600 to break the lock).
Which of these is best depends on various assumed probabilities, but if you have the calculation speed of a computer, the optimal wager is almost certainly either 6,589 or 2,911. If you are human, I'd still probably go with "either make your usual big wager or the minimum wager" mainly because everything considered in this post is too much to do in your head in those few seconds you have to decide on the wager.