Wikipedia:Reference desk/Archives/Mathematics/2015 November 26

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November 26

Most Disproportionately Popular

What's the name for the technique in statistics that was used to generate this map of the "most disproportionately popular cuisine" in each US state?

http://www.huffingtonpost.com/2015/01/14/most-popular-cuisine-state_n_6457252.html

Thanks! --50.35.66.30 (talk) 07:02, 26 November 2015 (UTC)

Unless they state which methods they used, we can't tell for sure, but to me it looks like they divided restaurants into a few dozen categories, determined the national average for percentage of each type, then compared each states's averages with the national average. For example, if 1% of all restaurants nationally are Taiwanese, but 2% of those in California are, then that's 200% the national average. If that was the highest percentage they found, then it wins for California. (I assume they had more significant digits than my example, BTW.) StuRat (talk) 07:48, 26 November 2015 (UTC)

The article explains what method it used! The categorizations were taken from Yelp, but otherwise it is as Stu says. But that isn't what we were asked. --70.49.170.168 (talk) 08:26, 26 November 2015 (UTC)

Thanks for your help. I do understand the method that was used to derive the results, I was just wondering if there's a concept or term in statistics for the subset of data points that are "most disproportionately x." Does that make sense? I looked at articles like statistical distance, for instance, but that didn't seem directly relevant. `--50.35.66.30 (talk) 09:32, 26 November 2015 (UTC)

The article didn't explain it in much detail. It said "it compared each percentage with the cuisine's representation in restaurants nationwide". But compared how, exactly ? That's where I made an educated guess, previously. Note that the method I described will tend to choose smaller categories of restaurants, as it's a lot easier for a state to have double the national average of a restaurant category that nationally only is at 1% than one that's nationally at 20%. Hence the rather obscure ethnic categories for many states, rather than the larger ethnic food categories you might expect, like Italian food. StuRat (talk) 07:30, 27 November 2015 (UTC)

Calculus in Fractional Dimensions?

I saw something about a college curriculum that I don't know whether it was a spoof or not. Within multivariate calculus, do you have the concept of Calculus over Fractional (like 3.5) dimensions?Naraht (talk) 14:50, 26 November 2015 (UTC)

Analysis on fractals is certainly a thing. Perhaps most fundamentally, fractals have a self-similar hausdorff measure, which allows functions on fractals to be integrated. Fractals are metric measure spaces, and so differential calculus is also possible in some sense. Probably the most important results concern the study of diffusion on fractals (e.g., percolation theory). But this all well outside of any standard "college curriculum" that one might teach to undergraduates in a multivariable calculus course. Sławomir
Biały 15:02, 26 November 2015 (UTC)

Fractal derivative is possibly relevant. Additionally, Lebesgue integration can be applied to the Hausdorff measure of a fractional-dimensional space. -- Meni Rosenfeld (talk) 15:12, 26 November 2015 (UTC)

Also, while this is a completely different thing than what you asked, I think an answer to a question that includes the words "fractional" and "calculus" should at least mention fractional calculus. -- Meni Rosenfeld (talk) 15:12, 26 November 2015 (UTC)

Is gambling sometimes worthwhile?

Is it worth sometimes to play a lottery? I know that they normally have negative expected value (in the same way as roulette, sport betting or horse races, among others).

However, what if a lottery has a jackpot that accumulates at each draw without a winner?

If the lottery has 1,000,000 numbers, $700,000 in prizes and each ticket costs $1, it is not worth playing without a jackpot. But how about buying 100,000 numbers in the same lottery with an accumulated jackpot of $2,000,000.--YX-1000A (talk) 16:17, 26 November 2015 (UTC)

Generally, as the jackpot goes up, more people play, so it never gets to the point where there's a positive ROI. This is especially true because the ROI is so low to begin with (lower than just about any other form of gambling). Also, once you figure in the tax implications, that makes the average ROI even worse. Then the advertised jackpot value is sometimes spread over many years, meaning you have to figure in inflation between now and then.

Also, buying more tickets does increase your odds, but at a decreasing rate. Consider if there were just 2 tickets sold, in a contest where tickets are drawn from a barrel, and you had one and somebody else had the other. Then you would have a 50% chance of winning. But if you spent twice as much and bought 2 tickets, then you wouldn't double your chances of winning, but only increase the chance to 2/3. StuRat (talk) 16:30, 26 November 2015 (UTC)

A gamble can be worthwhile even if its expected value is negative, if your utility function has a nonconvex region. Usually the utility function is convex, so the variance of a gamble adds insult to the injury of its negative expectation, but in some case it can be beneficial. This can happen if there is something you need badly, but can't afford.

Example: Your life-long dream is to travel to space. Your utility will increase by 1 unit if you manage to do it at least once in your life. Other uses of money will give you utility equal to the logarithm of your net worth (in $).

There are companies that will allow you to do this if you pony up the $$$ - $250K to be exact. But you can't afford $250K - paying this amount is either impossible or decrease your utility by more than 1.

If your net worth prior to paying for space travel is $X, then your utility without space travel is $\log X$, and after buying space travel, $1+\log(X-250000)$. So your optimal strategy will give you the maximum between these two values, and bends at $X=395494$. If you are close to this bend (to its left), you can increase your utility by gambling. For example, if your net worth is $395,000, then buying a ticket which costs $200 and has a 1% chance to give you $10K, will increase your utility by 0.000131. -- Meni Rosenfeld (talk) 17:36, 26 November 2015 (UTC)

• At least one lottery operator put this argument succinctly: "You can't win if you don't buy a ticket." --70.49.170.168 (talk) 22:18, 26 November 2015 (UTC)

• "...and you almost certainly can't win, even if you do." StuRat (talk) 08:04, 27 November 2015 (UTC)

The example I like to give is thus: "You owe X dollars to a loan shark, who is waiting outside the casino you are inside and is going to kill you unless you pay it back, in full, when you leave. The casino is about to close, you have less than X now, and nobody is willing to lend you money or buy anything you have. Thus, it's in your interest to gamble, with the best odds you can find, even if those odds are against you. (But for God's sake, if you manage to get X dollars, don't decide you are on a winning streak and let it ride !)" StuRat (talk) 19:35, 26 November 2015 (UTC)

See Run, Lola, Run. Or for that matter, Casablanca. --70.49.170.168 (talk) 22:18, 26 November 2015 (UTC)

If you don't account for enjoyment of the activity itself, spending $10 on lottery tickets makes more sense than spending $10 on a movie ticket, because at least you're not absolutely certain to lose money. If you do account for enjoyment of the activity itself, gambling is justified if you enjoy gambling, which as far as I know is the main reason people gamble in real life. -- BenRG (talk) 06:46, 27 November 2015 (UTC)

That argument is promoted by the gambling industry, but I don't buy it. I see that people would enjoy when then win money, but most of the time they will lose, and I don't believe they enjoy losing. Thus, there's a net loss of enjoyment. Why would they do it then ? Well, like any addictive behavior, they only think about the positives, and ignore the much greater negatives. (The junkie shooting up only thinks about the high, not the misery he will be in for later when the high wears off.) StuRat (talk) 07:20, 27 November 2015 (UTC)

Gambling is a special tax for bad mathematicians. Bo Jacoby (talk) 07:54, 27 November 2015 (UTC).

Stupidity tax. StuRat (talk) 08:01, 27 November 2015 (UTC)

The expected value is the likely averaged outcome if you play enough times for the the law of large numbers to apply - that would be enough times to achieve several wins and several losses. In the case of the lottery (unlike roulette), you will never be able play that many times, so the expected value is nor relevant to the question. --catslash (talk) 00:36, 28 November 2015 (UTC)

You seem to be misunderstanding what is the meaning of "expected value". Expectation is the single most important attribute of a random variable/distribution. There is a theorem of decision theory saying that, given a few reasonable assumptions, any rational agent can be modeled as maximizing the expected value of a utility function. Expectation is definitely not something which is only relevant to things that are repeated many times. -- Meni Rosenfeld (talk) 10:01, 29 November 2015 (UTC)

Expected utility is different to expected value I think. --catslash (talk) 16:37, 29 November 2015 (UTC) Are you referring to the expected utility hypothesis? --catslash (talk) 16:43, 29 November 2015 (UTC)

Expected utility is the expected value of the utility. "Expected value" is a general term that can be applied to any variable.

You may have been talking about "expected money". Expected money does not tell the whole picture, but if the expected money is negative it gives a strong indication that the expected utility is negative as well. So I certainly wouldn't call expected money irrelevant, either.

And yes, the von Neumann–Morgenstern utility theorem is the important decision theory result I referred to. -- Meni Rosenfeld (talk) 23:40, 29 November 2015 (UTC)

To answer the OP's question, it can happen and does, rarely. There was a lottery in Virginia (or a neighboring state), I believe, about 30? years ago that had badly drawn up rules and a prize that had accumulated. Some alert European syndicate bought up almost all the limited number of tickets and won the prize, making a few million in profit. It may have been noted in Harper's Index back then. That is my recollection, sorry I can't support it better.John Z (talk) 21:11, 28 November 2015 (UTC)

Maybe you're thinking of this case in the Massachusetts state lottery. A group of MIT students, for several years running, waited on building jackpots and bought significant proportions of the tickets and managed to make a lot of money. Staecker (talk) 23:58, 29 November 2015 (UTC)

No, I never heard of that case, I am pretty sure mine was around when I said - meaning I would be surprised if it was actually only 20 years ago. :-) Desultorily looked for it on the web, couldn't find it. In any case, more and much better evidence that this does happen occasionally.John Z (talk) 00:39, 30 November 2015 (UTC)