A brain teasing probability puzzle
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Hoss
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Thanks for the clips, I can't normally watch videos, but we're kinda slow today. They were interesting.
Did you notice that in both of those clips, it was nowhere near a doubling of chances? In the BBC documentary, the guy won 14 out of 20 cars, and the one who didn't switch won like 4. that's 20% / 70%. On mythbusters, switch won 37 out of 49 times, or 75.5%, while don't switch won 11 out of 49, or 22.4%. Those numbers show a strong advantage to switching, but it doesn't show a doubling of your chances as the math would indicate. It's more than tripling your chances.
Seems like something is missing in the theory.
You can run a simulator for a few thousand iterations and it'll get to 33/66. It is the nature of statistics that if you have a relatively small sample size it won't necessarily be exactly what you'd expect. After all if you ran the test once you'd have a 100% chance either way, based off that one data point. Statistics has a whole terminology and methodology for this.
Haast
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Exactly. This is why I ran 10,000 iterations in my simulation. You have to have a statistically significant sample to get proper results. I'm sure that is a larger sample than necessary, but it also runs nearly instantly on my computer, so 0 fucks given on oversampling.
Hoss, do you have a degree? What level and what field? And more importantly, did you ever have to take a statistics class?
Tuco
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I haven't watched deal or no deal because it's an awful show, but I imagine that this type of principle wouldn't apply in it because it's more of a continuous spectrum of reward instead of a bunch of goats and cars.
If there were 31 briefcases, 1 of which had a million and 30 of which had $1 and you get to the final two and don't discover the 1 million then yeah, you should switch.
But if there where 31 briefcases with value of 1million / i in them? I dunno.
If there were 31 briefcases, 1 of which had a million and 30 of which had $1 and you get to the final two and don't discover the 1 million then yeah, you should switch.
But if there where 31 briefcases with value of 1million / i in them? I dunno.
At the very end? Switch, if you are approaching it as a larger Monty Hall problem. The correct answer to a multi-stage Monty Hall problem is to stay until the last choice, then switch. But that assumes several things, up to and including that you didn't open the million dollar case already (which is not possible in the Monty Hall problem, you can't accidentally get rid of the top prize and keep playing). Also the bank periodically makes offers to buy you out... but, and this is key, the bank knows which cases are unopened (obviously, so do you) but itdoes notknow what is in a given case. It has no more idea where the million dollars is than you do. So the "host knows where the car is" part of Monty Hall does not apply to the bank's offers, it is going solely off the remaining total potential value in the cases. Which makes it like the "Ignorant Monty Hall problem" where Monty can reveal the car instead of a goat and screw you.
Deal or No Deal isn't a larger Monty Hall problem, as Tuco noted, and the differences makes the best choices different in some ways.
EDIT: Actually you can work out if you should take the deal or not pretty easily with a graphing calculator. Mine is dead atm, but I'll download an emulator of one and see if I can remember the steps.
OK so break out your handy graphing calculator and go stat > edit
Make a list in L1 with an entry for each value. You're set up to play the game.
Delete the values from the list as you pick them in the game (yeah, I know this is tedious if you're playing multiple times). Every time you get asked to make a deal, stat > calc > 1 var stats. If the average (X with the bar over it) is higher than the offer, no deal. If the average is lower than the deal being offered, take the deal. That'll give you the best average return for playing.
Make a list in L1 with an entry for each value. You're set up to play the game.
Delete the values from the list as you pick them in the game (yeah, I know this is tedious if you're playing multiple times). Every time you get asked to make a deal, stat > calc > 1 var stats. If the average (X with the bar over it) is higher than the offer, no deal. If the average is lower than the deal being offered, take the deal. That'll give you the best average return for playing.
I love this show. They don't always show the science, because sometimes the question being asked doesn't really implicate anything "scien-cy". And every once in awhile, there is just a myth they test that is absolutely mind-blowing.
My favorite myth is are elephants really scared of mice? When that was confirmed, I was hooked for life.
Hoss
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BS in Eletrical Engineering, and not that I recall.
How many iterations does take to get recognizable results? Does that mean those 2 tests are completely worthless?
Did your simulator pick randomly? If not, it was less than worthless.
Most statisticians, at a basic level, will tell you that a sample size of roughly 30+ will give you an accurate result to within a certain margin of error (though there are formulas for calculating acceptable sample sizes, given various criteria for specific situations that will be higher). But, over a large enough sample size, you will get the theoretical result. For instance if you flipped a coin 30 times you may or may not get 50/50, but it'd be close to 50/50 (there is a formula for calculating the acceptable margin of error). But over infinity trials, you will get 50/50. Realistically in most situations a few thousand iterations is sufficient to approach the theoretical result.
http://www.math.ucsd.edu/~crypto/Monty/monty.html
That might interest you. You have to play one at a time, but it saves the results of all players who have played. Unsurprisingly the results approach 66.6...% and 33.3....%
http://www.math.ucsd.edu/~crypto/Monty/monty.html
That might interest you. You have to play one at a time, but it saves the results of all players who have played. Unsurprisingly the results approach 66.6...% and 33.3....%
Haast
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Twins! I definitely took a stat class, though. With a decade past, it's a little hazy.
Good question. I would have to dig deeper to do the math for significant sample sizes. I just ran so many iterations that it would unquestionably be "enough". Were I sitting at my work computer, I could run more or less iterations and see how the outcomes were affected. The two live tests don't have enough samples to arrive at the theoretical values, but they are useful in showing the situation favors switching.
The simulated procedure is: put the car behind a random door and (separately) choose a random door. If Switch? is on, a goat door is revealed, then the unrevealed door is selected. Otherwise, the original choice is held.
The focus on empirical data is really odd. It's not like there are variables that are either uncontrollable or unaccountable. It's basic math. You can use a simulation or do the problem for yourself, but to delineate out theory and practice on a simple stats problem is asinine. What you're essentially saying is you think the foundations of math are wrong on a fundamental level, and once you're arguing that then I don't know what anyone can say.
Tuco
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Yup. It's fine to exercise with more complexity than is needed for fun, but one of the foundations of problem solving is to reduce the problem to its most basic components. Worrying about sample sizes for this problem is not doing that.
Hoss
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No, I'm saying that if the theoretical results don't pan out in the real world, then the whizzards of smart must have missed something. They are human, after all. Any theory that can't pass a practical test is a bad theory. You'd have to be a simpleton to trust what's written out on paper over actual results.
There are a million reasons why theoretical results don't match real world results and not a single one applies here. The "theory" that may or may not pass muster is, at its heart, 1 + 1 = 2. Do you understand why it's asinine to argue against that because of a lack of empirical evidence in the specific case of does one platypus plus one platypus equal two platypuses?
The theory does match the empirical results. You're just too suborn to see it. Your problem here is that you think 20, or 100, is a definitive sample size so you conclude that the theory doesn't match the empirical results, when it in fact, does. A coin flip has a 50/50 pecent chance of being heads, but that is not guaranteed over 1 flip, 10 flips, or even 100 flips. If you flip the coin 100 times and you get 53 heads and 47 tails, that doesn't mean that the theory doesn't match with empirical results. Given enough samples, any probability test will return to its true averages. That's why people play simulations of a million tests, to smooth out the variations. No one is going to sit around and play the monty hall game a million times.