I often get the gut-level feeling that a contestant who walks away on "Who Wants to be a Millionaire" is doing the wrong thing, so I decided to analyze the decisions mathematically and see what I can come up with. The first nontrivial level is the 2K one, so we start there.
My assumption will be that on average you have a 1/3 chance of getting a question right, up to the 50K level, when it drops to 1/4 because of the increased difficulty of the question.
2K, going for 4. Here you are risking 1K to win 2K, but we must factor in the additional chance of winning more than that. Thus, your true expected gain is 2 + 1/3x4 + 1/9x8, and I will stop there as this already gets us to about 4 as an expected gain. Hence you are risking 1 to win 4+, so you should always guess at the 2K level.
4K, going for 8. Here you are risking 3 to win 4 + 1/3x8 + 1/9x9 + 1/27x25. And again I stop here as we are at about 9 already, meaning you are risking 3 to win 9+, and should always guess here also.
8K, going for 16. Here you are risking 7 to win 8 + 1/3x9 + 1/9x25 + 1/36x50 + 1/144x150... This means you are risking 7 to win about 16, meaning you should go ahead and guess if you have it down to 2 choices. In reality this is often the case, and contestants often use their 50-50 here to get it down to 2 choices, but then they often walk away, which is a clear mistake as the odds say you should guess even if you have it down to 3 choices.
16K, going for 25. Here you are risking 15 to win 9 + 1/3x25 + 1/12x50 + 1/48x150 + 1/196x250. This means you are risking 15 to win about 22. The same comment applies here as at the least level. Contestants often get it down to 2 choices and then walk away. Mathematically, this is a clear mistake, although there are more legitimate arguments for walking at this level than at previous ones. It may be that the contestant really needs that 16K, but does not so much need the additional dollars.
25K, going for 50. This is a free guess so there is no walking away decision. However, there is another similar decision, which is whether to "walk away" from the question, since at this level you get the "switch the question" lifeline. IMO contestants too often switch the question at this level. It is a free guess, and I think if you can eliminate even one answer, then you should guess and save the lifeline for a future question.
50K, going for 100. Here you are risking 25 vs. 50 + 1/4x150 + 1/16x250 + 1/64x500. This adds up to 113, meaning the odds are clearly with you to guess, and you should always do so, even if you cannot eliminate any of the answers. Needless to say, contestants are much too risk-averse and often will walk away at this level when they should guess instead.
100K, going for 250. Here you are risking 75 to win 150 + 1/4x250 + 1/16x500. This is about 240, which again is more than 3 times the loss if you are wrong, so mathematically you should always guess, though as mentioned before more than math may come into play here.
For the last 2 questions the math is less of a factor, and other factors will determine whether you answer or walk.
Thursday, June 12, 2008
Monday, June 9, 2008
Should Big Brown Have Been 1-4?
Big Brown was 1-4 odds to win the Belmont Saturday, meaning you had to bet $4 to try to win $1 betting on him to win the race. While he certainly should have been the favorite, I questioned at the time why so many would risk money to win a smaller amount than they were risking.
In the wake of his last place finish on Saturday, I have reviewed the 29 past instances of situations where a horse came into the Belmont having won the first 2 legs of the Triple Crown. What I discovered is astounding. Of these 29 horses, only 11 have gone on to win the Belmont and the Triple Crown! This shows that the bettors who made Big Brown a 1:4 favorite had a grossly inadequate understanding of history.
It may also indicate something more sinister. To ask a horse to run a 1 and 1/2 mile race, especially when it is his 3rd race in 5 weeks, may constitute cruelty to animals. In light of past experience (11 horses in a row since 1978 that have won the first 2 legs then faltered in the Belmont), one must conclude it likely is animal cruelty and should be abolished.
In the wake of his last place finish on Saturday, I have reviewed the 29 past instances of situations where a horse came into the Belmont having won the first 2 legs of the Triple Crown. What I discovered is astounding. Of these 29 horses, only 11 have gone on to win the Belmont and the Triple Crown! This shows that the bettors who made Big Brown a 1:4 favorite had a grossly inadequate understanding of history.
It may also indicate something more sinister. To ask a horse to run a 1 and 1/2 mile race, especially when it is his 3rd race in 5 weeks, may constitute cruelty to animals. In light of past experience (11 horses in a row since 1978 that have won the first 2 legs then faltered in the Belmont), one must conclude it likely is animal cruelty and should be abolished.
Thursday, June 5, 2008
Wichita Smoking Ordinance Is Passed
The compromise smoking ordinance received final approval and will go into effect in 3 months. Businesses which want to allow smoking will have to pay a $250 annual fee and post a sign "no minors allowed". The medical community has railed against this compromise, saying it should have gone all the way and banned smoking in all public places.
Opposition to the ban has, surprisingly to me, come not from smokers who want to be able to go into a bar or restaurant and smoke, but from folks who fancy themselves to be private property advocates. To these people, a private property owner should be able to do anything he wants with his property. This of course has never been the law. Inside the city limits of a city like Wichita, there are all kinds of limitations on what one can do with his property. Health Department and city inspection regulations come into play, and ultimately the city can take your property and condemn it if you refuse to bring it up to the city's standards. As George Costanza once famously shouted, "We live in a society here!"
And when you are using your property to run a business which holds itself out as being open to the public, many more regulations come into play. Restaurants, for example, are subject to unannounced inspections and can be ordered to clean up their operation if deficiencies are found. By the reasoning of these private property fanatics, there should be no Health Department regulations of restaurants because that infringes on private property rights. By their argument, restaurants should be able to sell contaminated food, and once word gets out people can then go elsewhere. That is their pro-smoking argument, that each business should do as it pleases regarding smoking and the marketplace will decide who survives and who fails, and public health concerns be damned.
Opposition to the ban has, surprisingly to me, come not from smokers who want to be able to go into a bar or restaurant and smoke, but from folks who fancy themselves to be private property advocates. To these people, a private property owner should be able to do anything he wants with his property. This of course has never been the law. Inside the city limits of a city like Wichita, there are all kinds of limitations on what one can do with his property. Health Department and city inspection regulations come into play, and ultimately the city can take your property and condemn it if you refuse to bring it up to the city's standards. As George Costanza once famously shouted, "We live in a society here!"
And when you are using your property to run a business which holds itself out as being open to the public, many more regulations come into play. Restaurants, for example, are subject to unannounced inspections and can be ordered to clean up their operation if deficiencies are found. By the reasoning of these private property fanatics, there should be no Health Department regulations of restaurants because that infringes on private property rights. By their argument, restaurants should be able to sell contaminated food, and once word gets out people can then go elsewhere. That is their pro-smoking argument, that each business should do as it pleases regarding smoking and the marketplace will decide who survives and who fails, and public health concerns be damned.
Sunday, June 1, 2008
Final Jeopardy Wagering Strategy--Part 3
A striking example of violation of what I call the Law of Separation occurred this week. It was in the show which is showing old games, on at 4:00 P.M. here.
The player was Mehrun Etebari, who had won 5 in a row and was playing in his 6th game. He trailed the leader $13,200 to $17,200 going into Final Jeopardy. He then made the horrible wager of $6,700! As soon as his wager was revealed, he muttered, "bad wager", and he was right. The leader wagered $9,201, as expected, and then hung on to win, even though both missed the question, because Mehrun had wagered too much. He had to wager $5,200 or less, creating the "separation" of at least the $4,000 difference in their scores so that he would win if both missed the question.
The player was Mehrun Etebari, who had won 5 in a row and was playing in his 6th game. He trailed the leader $13,200 to $17,200 going into Final Jeopardy. He then made the horrible wager of $6,700! As soon as his wager was revealed, he muttered, "bad wager", and he was right. The leader wagered $9,201, as expected, and then hung on to win, even though both missed the question, because Mehrun had wagered too much. He had to wager $5,200 or less, creating the "separation" of at least the $4,000 difference in their scores so that he would win if both missed the question.
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