Thursday, March 30, 2006

Guess the Fraction of White Cue Balls

IMAGINE that you are a contestant in a Game Show whose prize is a cool $1 million.

An Olympic-sized swimming pool is filled with a pre-arranged mixture of regulation billiard balls (colored and numbered 1 to 15) and white cue balls. The object of the Game is to guess how many white cue balls there are. It has been announced that the TOTAL NUMBER of balls whether white or colored is exactly 100 million.

Contestants are given exactly one hour during which they can perform any physical, mental or statistical inspections on the collection of billiard balls. They are allowed pencil, paper and a computer with an Internet connection. Rulers, clocks, weighing scales, cameras, cell phones are also available. The winner is the contestant who comes closest to the correct answer in one hour.

What do you think is the best strategy to win the $1 million prize?

[NOTE: There is no "correct" answer, but participating in this exercise could change the way you look at and interpret public opinion surveys such as the recent Pulse Asia Inc. Ulat ng Bayan national survey on charter change, or this one on political scenarios. ]


Econblogger said...

Do I also get a bucket?

Rizalist said...

Haha, anything you can carry yourself into the contest area is allowed, as long as it is a readily available commercial item.

Marcus Aurelius said...

Calculate the volume of the pool and the bucket.

Start to take dig out bucket fulls of balls. Count the number of cue balls in each bucketfull and repeat each time writing down the number of cue balls obtained by a bucketfull. Figure out the average number of cue balls in each bucket and then extrapolate from the average number of cue balls obtained by the bucket to come up with a guess on the total number.

One may have to adjust their guesses as we all know pools have varying depts so to calculate the volume based on the product of length, width, and height would not be accurate. Perahaps one would divvy it up into three zones the shallow, the deep, and the transition and apply volume formulas appropriate for each zone.

Rizalist said...

(INFO 1) It has just been revealed that the TOTAL NUMBER OF BILLIARD BALLS IN THE POOL IS exactly 100 MILLION.

(INFO 2) A typical contestant in previous shows takes ONE SECOND to count one billiard ball. So a completely accurate answer would require a minimum of 100 million seconds, or about 3.1 years of just brute force counting. There are only 60 secs/min times 60 minutes/hour or 3600 seconds available to the contestants.

So Marcus, what are you suggesting to do during the one hour?

You could take a single 3600 ball random sample, noting the number, call it NWHITE of white balls. Then an estimate for their total number would be NWHITE/3600 times 100 million.

But is this the best we could do?

arlan said...

before you mentioned 1 million balls i was batting for 1/16.

now i'm going for a range from 0 - 1/16.


arlan said...

i take that back, i'm still for 1/16, hehe

Rizalist said...

Arlan--The total number of balls is known to be 100 million. The fraction of white cue balls could be any number between 0.0% and 100%.

It is analogous to the UNKNOWN percentage of people who are for GMA to resign for example, or the number of people who want her NOT to resign.

We are interested in the best strategy for doing that given the constraints of the game show example.

The organizers did not put in complete sets of colored billiard balls either, so the number of 1s, 2s, 3s, etc are also unknown to all but the game show organizers.

Rizalist said...

So far we have one definite strategy -- let's call it the Aurelian strategy for Marcus Aurelius who just suggested it.

Here the contestant spends one hour counting out 3600 balls at random, noting how many of them are white balls. Suppose for convenience one contestant counts 1200 whites out of the 3600 total random sample. His estimate for the fraction of white cue balls would therefore be 1200/3600 or one-third, leading to an extrapolation that there are about 33,333,333 white cue balls in the swimming pool.

How accurate would such an estimate be?
We know the answer. It comes with a very fancy name: the Normal Approximation to the Binomial Distribution and boils down to a simple rule of thumb. The Margin of Error in such a statistical experiment is roughly equal to the reciprocal of the SQUARE ROOTF of the random sample, in this case 1/sqrt(3600) or plus or minus one sixtieth. So, using the Aurelian Strategy, we get a statistical "survey" result of the billiard balls if one third of our 3600 ball random sample are white cue balls:

33,333,333 plus or minus 1,666,667 whites cue balls OR

33% plus or minus 1.67%

Note that the Pulse Asia surveys samples are 1200 respondents. If you compute the reciprocal of the square root of 1200 you get the famous Margin of Error in SWS and Pulse Asia Surveys of 2.88% (which is often rounded to plus or minus 3% in reportage).

This gets a lil deeper if you wanna understand that bit about 95% confidence at 90% probability.

But if you were a contestant, is this really the best strategy? A single 3600 random sample?

AmericanPainter said...

Gosh! I hate math and use none in my art work. I would just use the cell phone to call my old college math professor. He would miss the correct answer by a mile but I would have missed it by twenty miles!

This pretended contest takes us away from the current subjects at hand. A ploy used successfully by Gloria. Rizalist, you’re learning!! - LOL

Jon Mariano said...

I suggest:
1. Select 9 points of 1 square meter each in the pool, maybe tic-tac-toe points.
2. Count the number of cue balls on top (visible).
3. Measure the depth of the point, divide by the diameter of the balls, then multiply the two. Note the resultant product for each point.
4. Get the average for all the points.
5. Multiply the average with the area of the pool.

Rizalist said...

AP--this wasn't actually intended to be a math lesson at fact it's about the art of ... lying!

JON -- nice try, i guess you could take photos of the ensemble and may be couunt the visible balls on the outer surface. But nothing actually beats the random sampling approach.

Marcus Aurelius said...


A few things on your approach. This is how crowd sizes at gatherings are estimated. Photos are snapped of the crowd in such a way the area the photos encompass is known. Then the total area of the mass of people is figured out. They take those photos, count people, and based on the average # of people and the area of the crowd and the photos calculate the total number of people in the crowd.

Both your approach and my approach depend on a uniform distirbution of cue balls throughout the depth of the pool. This is why we take multiple samples, in the hopes areas & buckets that oversample cue balls will eventually be offset by undersamples in others. Wow, I could really go on about this.

The upshot is a thoughtfully constructed poll and just as thoughtfully conducted can get close to the actual result.

I'll continue discussin at Blogger Beer.

Jon Mariano said...


Can you imagine using a bucket in an olympic sized swimming pool? How deep into the pool are you going to use the bucket? What are you going to do so that the balls will not cascade?

I chose the linear way (2D) way of counting the samples because it's going to be very difficult if done in 3D.

The more points of sampling would make it accurate I guess...

Jon Mariano said...

When these methods are applied to surveys, we can see how un-scientific they really are.

arlan said...

uhm, if i win, would the $1M be paid in full or installment? cash? because i'd be issuing post-dated checks for the 1000 or so people i would hire to do the counting. oh, and to pay for the same number of buckets, too.

Marcus Aurelius said...


Well, there is no digging in much below the surface, however taking a bucket from the top gets you a volume computation which is easier than basing it on sampling an area and then essentially "integrating" that surface throughout the volume of the pool.

Also, this is *_EXACTLY_* how surveys are done.

Over at Blogger Beer I discuss this discussion and perform some more analysis. In addition, I pose the following experiement on randomness.

Go to the grocery store (head to SM's grocery store or any that totals the prices closely) and buy some groceries. While you place your items into the cart round off the price to the nearest quarter (or 25 centavos or to any convenient peg) and keep a running total of the rounded prices. Now, with ten items, you could be off by +/- $2.40 (.24/item is the maximum error here, or whatever currency you use). However, typically the price you pay is very close to what you totaled up in your head, and the more items you purchase the more likely your total will be close to the cashier's total.

Why? Because, randomness means you will run into about as many items 2 cents over a quarter as you will run into items 2 cents below the quarter; the extremes on both sides cancel each other out.

The science of polls is very well understood, and the abuse of polls is very common. This is why when a poll comes out you will hear people urging you to look at the poll's internals. For example, it would be an abuse to take a poll on President Bush's popularity in Berkeley California and then claim the results are representative of America as a whole.

A thoughtfully constructed and conducted poll can be a very accurate measure of a population's characteristics. In WWII statistical methods predicted with extreme accuracy German war production and in one instance was 100% accurate (sorry, this comes from memory of a story a buddy in his second semester of statistics related to me, will search it out and report on Blogger Beer). Statistics is a very powerful field of mathematics.

Rizalist said...

Thanks for that Marcus. BTW, hope you do visit SWS and Pulse Asia more and check out the questions and surveys. They really do suffer from abusive and invalid practices. And it never ceases to amaze me how the task of exposing exactly what is happening in a given survey never ends itself. There is extreme creativity in the use of statistics to spread falsehoods and misdirections.

For example, my reading of the latest Pulse Asia surveys is that 67% of the people actually support Chacha now or soon.

Marcus Aurelius said...


Reading the latest Pulse Asia poll, I not feel comfortable concluding 67% of Filipinos want cha-cha now or in the near future.

According to the language in their press release "No, the constitution SHOULD NOT BE amended now, but it may be amended at sometime in the future" gives a lot of wiggle room. If the question they use to pigeon hole respondents is the same or very similar I would not categorize them together with the "Yes change it now" category. It may not be completely appropriate to lump them in with the cha-cha oppositionists but their position is closer to opposition than to support. Furthermore, it is unpredictable as to how that 24% will break in the future. It is clear as to which position relies on the "wait and see" group.

That 24% is the crowd the anti-cha-cha and pro-cha-cha (driving around one day we went by Antipolo, I asked where Propolo was it took everyone a bit to get it) need to play to. Hehehe, a relative of friend is on some mayoral staff in the Bisayas and was approached by her boss to support cha-cha. She thought about it and adamantly said no. Fortunately, her boss is respecting that decision.

I wonder what sort of things would influence the 24% to more definitively support one side or the other?