Probability: The Menu Problem

The first thing to do is to objectively grade yourself in the social pecking order. 1 to 10 is too rough and ready a guide for a question of this subtlety. 1 to 100 would be more calibrating.

From the tone of your post and your obvious earnestness, I'm assuming you're not taking the piss out of us, a dangerous thing to do but I'm nothing if not daring in these matters, I would suggest the higher 30s although your literary style leads me to think I might be flattering you a little, it might help if you studied the prices, decor and servility of service. The food is neither here nor there.

Mark those out on the same scale.

Without giving the matter a lot of thought I think the soup might well provide an easy and resonably accurate guide. It has all come out of A4 catering tins supplied by the various manufacturers anyway and it's all more or less the same price at the backdoor.

The chef in any particular restaurant might add various items to produce a distinctive "homemade" sensation and to reflect his individuality. A plastic cup full, with a lid, from a fish and chip shop would be about 95 pence ( 94 cents on Friday if I have my sums right), a bowl of it with a roll and butter would be about £4 in a beta-minus establishment (that's one where they clean the fly **** off the bulbs once a week) and from there upwards through The Cotswolds in the tourist season, the Cafe Royal and the Royal Enclosure at Ascot in the "strut your bank account" festival in June where astronomical prices are asked and paid.

It is being seen in these various places which is the essence of the matter. It is where you chomp through the nutrient bed that is of importance and not what you're chomping on. The latter is here today and gone tomorrow, as they say of things in transience, but the former will be remembered by your peers and your betters for considerable periods of time.

What you should do is aim a few points on the scale higher than the position you allotted yourself in the pecking order. Never lower. That is crass vulgarity. Where people know you I mean. If you are anonymously travelling bargain nutrient is what to look for as this frees up resources, which, as Mr Galbraith might have said, are always scarce.

What is supposed to happen, although it isn't guaranteed, is that if you follow this simple procedure, cutting down on other things such as smoking and drinking if necessary, or taking out loans, you will insensibly find yourself rising up the pecking order to match the grading of the soup price. That is a somewhat mysterious process which would take too long to analyse in a short post such as this one is intended to be.

You then raise your sights and if it keeps working you eventually need a dump-truck to carry your head around in.

There was a chap once who slept in the back of his car, abluted in the public conveniences and denied himself all sorts of luxuries in order to dine in the Cafe Royal in Regent Street and he became the editor of a great national newspaper and was feared throughout the land for what scuttlebuck he had a record of in his safe.

I thought it polite to provide you with a response as you have thanked us in advance and one naturally feels that having been thanked one ought to do something to be thanked for otherwise you could look a bit of a mutt.

Thanks for keeping it brief, Spendi...

:wink:

My answer is 3.25

Your over-rapid response Rocky, and I sincerely hope over-rapid responses are not habitual, has prevented me from improving my essay by inserting an "a" into reasonably, as I have just done, and thinking that "in transition" is preferable to "in transience" as it has fewer intellectual connotations.

I just woke up, S.

Please pardon my intrusion into your art.

I'll go back to reading along from the gallery.

(welcome to A2K talexndr, and there are many folks here nicer than Spendi and I)

Rock

Re: Probability: The Menu Problem

Talex -- before I answer your question, let me make sure I understand how you mapped your real-world problem onto your math. As I understand you,

• You have a new restaurant with N items on the menu.
• Each time you sample one item, the outcome is either +1 for "like the meal" or -1 for "dislike" the meal. That way, if the restaurant has as many good meals as bad ones on the menu, the restaurant's score is 0.

If this is an accurate description of your experiment, here are the points I don't understand:

• What's your statistical definition of "like the restaurant"? That you'll like every item on the menu? Maybe 80% of the items on the menu? Maybe 51%? Whatever your definition is, you need to have one.
• Where would a standard distribution come in? In the model I just described, you have two values, 1 and -1, each with its own probability. That's not a standard distribution. Maybe I misunderstood your model?

I'm looking forward to hearing back from you. It's an interesting application of statistics.

Menu Problem Clarification
Thank you all for your posts, both colorful and statistical

Here is more elaboration to the Menu Problem:

While we might apply the like/dislike binary simplification, black or white does really apply so well to actual experience. The beauty of the normal curve is it allows shading (literally and figuratively). In the Menu Problem, the mean and the 10% to either side of the mean maps to the subjective experience of "best quality present." And because the curve and it's error are normal, we need not quantify what "best" really means to the restaurant or ourselves, nor any curve overlaps between our preferences and the restaurants offerings. Our minds will map those automatically.

Of course the best dish we sample at the restaurant may be an outlier! But thanks to the normal curve, we can be more certain that we did not, especially since I am assuming we try more than one dish. The chance of hitting an outlier three times is slim (my 3 Strikes Guideline). Possible but slim.

So this is not a problem in determining the mean quality of the restaurant -- which would then turn the Menu Problem into a sample size issue. Rather, how many dishes would you have to sample to be 80% confident that your most favorite meal eaten there was within 20% of the mean quality of the restaurant? In other words, how many dishes do I have to try before I confidently decide I like the place, or conversely give up and try another restaurant.

Using the Monte Carlo paradigm, which might be a great Excel project if I had a better grasp of the math, how many darts would I have to throw to be 80% sure that ONE of those darts is within 20% of the shaded area around the mean (10% to either side)?

Ultimately, the problem's solution should be generally applicable to any of the "menu" issues we always bump into. How many tracks of a 10-song CD should I listen to before I can safely decide to buy or not? How many times should I try a particular route to work before I know whether to keep that course or not? How many French women do I have to date before I give up and try Italian?

Thank you all for all your kind efforts in this long term quest of mine!

Gratefully,
Don Quixote

Are you sure the dart analogy fits?

Determining how many darts have to be thrown to be 80% certain that one of the darts lands in the shaded region is different than determining how many darts have to be thrown to be 80% certain that a particular dart (your favorite meal) is in the shaded region.

Dart's Analogy
Excellent point!

It is the first case: How many darts do you have to throw to be 80% certain ONE of the darts lands in the shaded area?

We are not trying to find "our subjective favorite meal." Rather we are trying to discover at least one meal around the mean of the restaurant. We would subjectively know which meal we enjoyed the most and least and their qualitative ranking. Though we would NOT know which of those meals/darts is the one landing around the mean. It may have been our favorite meal there or our least favorite meal there.

Ultimately then, the problem becomes a "risk issue." Confident that one of your meals/darts was around the mean, the highest risk would be assuming your best meal was the one around the mean and thus call the restaurant a real winner, spending more money there. Going back again and again until your mind is changed or more confident. The lowest risk would be assuming your worst meal was the one around the mean and not spend any more money at that restaurant, trying a new restaurant.

With thanks and a smile,
alexander

Re: Menu Problem Clarification

I'm sorry to be a party pooper, but it doesn't work this way. Before you wax eloquent about any particular curves, and about any particular properties of these curves, you have to describe the experiment whose results these curves describe. So if I may be so dull as to repeat my questions: What constitutes a datapoint in your statistical curve? Are you going to a restaurant, order a random item on the menu, and, after tasting it, give a thumbs up (+1) or thumbs down (-1)? I'm guessing that's what you do, but I'm not saying it anywhere. 2) How do you define "like the restaurant" / "dislike the restaurant" in terms of your experimental results?

I think it's possible that the nail in the "menu problem"'s coffin is that restaurants have incredible variance. As a former food service employee, I can tell you that even individual dishes can vary wildly from one preparation to another, not to mention the quality you could expect from every dish at a restaurant.

Gee Vengo-

I hadn't realised that you had been a food service employee.

Were you working your way through a degree in science?

Were you a washer up?

Menu Problem minus the menu
My apologies for the confusion. Perhaps it would serve best if I restated the question more simply without the additional explanations.

Given a normal distribution with mean = 0 and SD of 1, how many samples would I have to take to be 80% confident that at least one of the samples (any one of them) falls within 20% of the mean (10% either side of the mean)?

Explanation of the solution so I might understand it would be helpful. Excel Monte Carlo method would also be helpful.

Thank you everyone!

I would just eat what is put in front of you and be thankful you have it for so little effort and stop fussing about like a finnicky princess.

It's a Math Problem
Spendius,

Like all poorly raised children, you talk alot but say nothing. And when your babble goes ignored, you become a temper tantrum of pettiness.

There there. Are you happy now? One of the adults noticed your screaming.

It's a math problem with application to dozens of real-world circumstances. Perhaps it would serve everyone best, including yourself, if you pretended to be of significance in a place where your insignificance were not so laughably obvious.

Here we do math. Try it!

With best regards in your search for importance,
t

This thread is a hoot.

But who woulda thought?

Why have just a +/- rating for each dish? Even if you sorta kinda liked everyone of the items on the menu, you still might never go back if there wasn't at least one that was outstanding, a 9 on a scale of 10, let's say.

That is not to mention the service, cleanliness, etc. I believe the math might be incredibly complex and still be likely to mislead.

I can't do tantrums as good as that. It really thrummmmmmms with indignation.

Surely the cost of restaurant meals is proportional, directly or inversely at discretion, to the sense of self importance or humiliation purchased.

As a mere male I stay well away from them.

Truffaut says in The Man Who Loved Women that restaurants are threatening places for men. That danger lurks therein.

I think the solution to your problem is common sense. And it depends on how hungry you are. As Socrates taught.

talex-

I can't do tantrums as good as that. It really thrummmmmmms with indignation.

Surely the cost of restaurant meals is proportional, directly or inversely at discretion, to the sense of self importance or humiliation purchased.

As a mere male I stay well away from them.

Truffaut says in The Man Who Loved Women that restaurants are threatening places for men. That danger lurks therein.

I think the solution to your problem is common sense. And it depends on how hungry you are. As Socrates taught.

Whoops! This hemlock is making me feel funny.

Too late to remove your double post which will now haunt you till eternity. ..