Infinite vs Infinite

November 21, 2006 1:35 PM

ProbabilityFilter: Calling all mathematicians...

I am not a math person, but this is something I've always wondered/had trouble wrapping my head around:


Imagine you are standing on a football field, there is a trash can about 30 yards away, and you are holding an average sized stone.

I submit that there are _infinite_ ways (different trajectories) of tossing the stone such that it lands in the trash can.

I also submit that there are _infinite_ ways of of tossing the stone such that it doesn't land in the trash can.

Surely the probability of missing is higher than making the shot? Yet if both are infinite, are they not the same probability? Are there not, in fact, infinite ways to achieve the former? Or is there something else I'm not getting?


Extra Credit: Does the math change at all if I'm _trying_ to make the shot vs. randomly throwing the stone in any direction?

posted by 777 to science & nature (26 comments total)

Are each and every type of throw equally likely: no.
posted by chunking express at 1:41 PM on November 21, 2006

Does the math change at all if I'm _trying_ to make the shot vs. randomly throwing the stone in any direction?

Yes, obviously, it changes enormously. Your question is unanswerable without making assumptions about the distribution of the shots; counting the "number of ways to make the shot vs. not make it" is meaningless when both of those numbers are infinite.

That's a trap that confounded a lot of the pioneers in probability theory and led to apparent paradoxes such as you seem to be struggling with at the moment.

But putting down a technical definition of what constitutes a "distribution" in this context is probably irrelevant. You know very well that, for a well-practiced person with precision aim, the probability of the stone going in the can could be very high indeed; for a blindfolded, inebriated hooligan facing the wrong direction it's going to be a lot lower. The characteristics of the thrower determine which areas of the field will receive a large number of hits (over repeated trials) and which areas will remain relatively devoid of hits.

If you picture many, many trials of the same thrower plotted with little dots on a diagram of the field (to show where the stone landed), you'd have a picture where the density of dots shows what areas are likely to be hit. (That picture is a graphical representation of the concept of a "distribution" that I'm talking about).

And you can think about this: If you do that experiment a large number of times, you can estimate the probability of a hit by computing (# of hits) / (total # of trials). That'll give you some number in the range 0 to 1. As you repeat the experiment over and over, doing more and more trials, both numbers in the fraction get large - you're right about that - but the quotient which represents the probability of a hit will stay settled near some number between 0 and 1. You're not interested in "dividing infinity by infinity"; you're interested in looking at the long-term behavior of a fraction as both its numerator and denominator get large.

Are each and every type of throw equally likely: no.
That's a little misleading if it suggests that individual points (where the stone could land) or trajectories (that the stone could follow) have a meaningful, nonzero probability of occuring in a problem like this.
posted by Wolfdog at 1:58 PM on November 21, 2006

Let's transfer this to the real line, the set of all real numbers. Let's say that hitting is equivalent to randomly selecting a positive integer.

What's the probability of a hit: zero, because for every hit there are an infinite number of misses, even though there are an infinite number of ways to score a hit.

This isn't really a probability question. The probability approach would be to note that randomly thrown rocks have a certain probability of landing inside a given area, and that probability is most likely very close to (area of trashcan) / (area of field).
posted by ROU_Xenophobe at 1:58 PM on November 21, 2006

Unfortunately, having an infinite number of choices does not lend itself well to probability.

Narrow the problem a little. Assume a 2-D case where all you have to pick is the angle and velocity of the initial trajectory (this is a good high school physics kinematics problem). If you have a level field, then your angle choices range from 0 (horizontal) to 90 (vertical) degrees. There are an infinite number of angles within that set (30, 30.1, 30.11, etc.).

The probability of picking the "right" angle for a given velocity is 1/infinity, which equals zero. You have zero chance of picking the proper angle. Using the same logic, you also have zero chance of picking a wrong angle.

So, yes, you are equally likely to pick a correct trajectory as an incorrect one. However, since they're both zero, this answer is meaningless.

The major flaw in your thinking is that statistics needs a finite set to do the math on, otherwise you get these divide by infinity errors.

This was fleshed out (rather humorously) in The Hitchhiker's Guide to the Galaxy. There are a finite number of people in an infinite universe, therefore the population of the universe is zero.

If you want a meaningful answer you need to simplify. You can only pick integer angle values, for example - say, with a ball-throwing machine that has a bunch of stops on it.

Sorry for the long-winded answer.
posted by backseatpilot at 2:02 PM on November 21, 2006

...And I need to learn to type a bit faster, apparently.
posted by backseatpilot at 2:03 PM on November 21, 2006

Infinite doesn't mean "a really big number." There are a lot of different ways that a stone can be thrown, but not infinite. Maybe the number has something to do with the density of atoms or the size of an electron or something else, I don't know IANAS, but there's a physical limit.

The limited number of trajectories that make the shot are smaller than the total number of trajectories.

So this isn't a case of infinity / infinity, it's a case of REALLY big number / REALLY REALLY big number.

A good way to estimate that ratio would be to measure the angle that would make the shot and divide that by 360.
posted by revgeorge at 2:05 PM on November 21, 2006

The probability approach would be to note that randomly thrown rocks have a certain probability of landing inside a given area, and that probability is most likely very close to (area of trashcan) / (area of field).
Well, again, not unless you're assuming a specific kind of distribution of the throws (which is unlikely to match anything generated by a human being even if she's trying to distribute them uniformly over the field.)
posted by Wolfdog at 2:05 PM on November 21, 2006

What ROU_Xenophobe said. There are an infinite number of points on a dartboard, but that doesn't stop me throwing a bullseye. (My total lack of skill does though)

You could define a 6 dimensional space made up of the initial position of your stone (x,y,z) and its initial velocity (v_x, v_y, v_z) (and you might consider adding further dimensions representing its initial rotation and rotational velocities, but we can ignore those for now).

Each point in this space represents a throw, and some of these will hit and some will miss. If you choose some volume of this space to represent the limits of the problem (so that you don't throw the stone inhumanly fast, throw it backwards etc. and just limit it to throws you might actually do) you can use the ratio of these six dimensional volumes to work out your probability.

If you really wanted to be accurate you'd add more things to this phase space, and maybe weight points by how likely you are to throw them etc., but the general principle of looking at the volume of points rather than the number of points in a volume (which is always infinite) should help you see how to do this I think.
posted by edd at 2:08 PM on November 21, 2006

I'm not entirely sure there's more than a suggestion that it's not really infinite as revgeorge says. I'd want to have a working quantum theory of gravity before committing to that.

It might be, but it might not be, and it being infinite doesn't stop you answering the question.
posted by edd at 2:11 PM on November 21, 2006

if both are infinite, are they not the same probability?
No. Infinity is not a number, it's a concept. Not all infinities are equal.
Consider the ratio x/2x. Now make x infinitely large. Both the numerator (x) and the denominator (2x) are infinite, but the ratio of the two is still 0.5.
posted by rocket88 at 2:11 PM on November 21, 2006

All infinities are not created equal, and you can still deal with probabilities when they are involved.
There are an infinite number of integers divisible by 3.
There are an infinite number of integers not divisible by 3.

But if I pick a random integer, I'm twice as likely to get a non-divisible-by-3 number than not.
posted by 0xFCAF at 2:15 PM on November 21, 2006 [1 favorite]

Uh, backseatpilot, infinite numbers of choices are just fine. That's what continuous probability distributions are. The samples will be finite, naturally, but the space of possibilities doesn't need to be.

The step you're missing is something that is, in math, called "measure." It is possible to sum an infinite series of numbers and get a finite result if the values in the series get smaller and smaller quickly enough. When you are tossing the ball at the trash can, you will have a certain distribution of locations your ball will go. Probably a lot of them will be close to the trash can and fewer and fewer will hit farther and farther away. If, somehow, all of your throws landed on a line with the trashcan, it might look something like one of these curves. Let's say the trash can is between x=1 and x=2. If we want to know the probability that the ball will land in it, we can just find the area under the curve between x=1 and x=2 and we get a measure of the probability that the ball will land there. Since the total area is finite (and, for probabilities, is multiplied by a constant so that the total sum is equal to one by convention), we now have a way of comparing things that are both infinite (the number of ways for a ball to land in or out of the trash can) in a reasonable way. It is this initial curve of where the ball might land (which is called a "probability distribution") that will change based on how you are aiming, and thus the end probabilities and math will differ.
posted by Schismatic at 2:17 PM on November 21, 2006

For a nice, not too hard to follow illustration of the crucial importance of defining what you mean by 'choosing at random' ahead of time, check out this discussion of Bertrand's paradox at Wikipedia. You will note that infinite sets are involved in all branches of the problem , and that the arguments remain perfectly valid despite that.

Once you've done that, your problem will have a well-defined, but complicated solution, in my opinion.

However, defining what your random variables 'should' be requires a great deal of argument about the physics of the situation at each one of quite a few steps for the problem you've chosen, I think, and after you're done, I believe most people would say a lot of scope for disagreement remains.
posted by jamjam at 3:20 PM on November 21, 2006

I choose not to believe in infinity. Nobody's ever showed me evidence of any infinity and it has never served any practical purpose. It's a structural invention of thought. Don't trust anything in life you couldn't write a computer program to solve.

LET maxV be your maximum throwing speed
LET maxIter be the number of trials your computer can make within your attention span of this problem
LET s:=0, f:=0

LOOP (i=; i< maxiter; i:=i+1)
[
CALCULATE random angle, a in (0 .. 90)
CALCULATE random velocity, v in (0 .. maxV)
LAUNCH stone
IF stone lands in trash can according to laws of physics INCREMENT s
ELSE INCREMENT f
]

OUTPUT s/(s+f)

That's all you will ever need and the most you can ever trust.
posted by zaebiz at 5:22 PM on November 21, 2006

Schismatic is right. This deals with measures, i.e. continuous probability distributions. A good way to think about this is that continuous distributions measure how likely you are to be 'close' to the target, in a precisely defined way.

The extra credit problem actually gets a little more to the heart of the original issue. To say that you are trying to make the shot indicates that your probability distribution isn't uniform. To determine the probability that you make the shot, you must first specify the distribution function, then integrate over the target area to find the answer.

As an aside: if you don't know what an infinite cardinal or ordinal is, please refrain from writing things about infinity. All you're going to do is confuse people with your rambling nonsense. The only people qualified to answer this question are those who have some rigorous understanding of probability/measure theory.
posted by number9dream at 5:24 PM on November 21, 2006

What Schismatic and number9dream said.

backseatpilot,

The probability of picking the "right" angle for a given velocity is 1/infinity, which equals zero. You have zero chance of picking the proper angle. Using the same logic, you also have zero chance of picking a wrong angle.

Schismatic addresses this, but let me put it another way. You're correct, but only because one of your assumptions is wrong. You're assuming there's only a single angle that for a given throw velocity will hit the wastebasket. This is only true if the wastebasket is a point and the rock is a point. In real life, the wastebasket has a finite area > 0 which means there's a range of angles that'll hit it for a given velocity. This is why you can look at the areas of the wastebasket/non-wastebasket world in conjunction with the throw probability distribution and get the probability of hitting the wastebasket. Similar logic applies if you want to figure the probability of covering a point with a non-zero area stone.
posted by Opposite George at 5:52 PM on November 21, 2006

You're over-thinking this.

I submit that there are _infinite_ ways (different
trajectories) of tossing the stone such that it lands in the trash can.

Wrong. The stone and the trash can are fixed sizes and the trajectories must resolve to one set of conditions -- that is, a trajectory where the ballistic angle and gravity combine to allow the stone to fall into the can.

The number of this set is extraordinarily large, but it is by no means infinite.
posted by frogan at 6:46 PM on November 21, 2006

Guess I should have paid more attention in my stats classes. I knew there was something wrong with my argument when I was fleshing it out, but I couldn't put my finger on it...

Maybe I should stop reducing my finger to a point-mass.
posted by backseatpilot at 7:07 PM on November 21, 2006

The first question, without aiming, is really about cardinality. Countable and uncountable infinities, and so much more.. The second question is mostly basic calculus.

The one dimensional analogy, ala ROU_Xenophobe, is all you really need to understand this better. You are making a random toss at the real numbers, and you are trying to hit a particular range, say between 0 and 1.

So, the real problem is now a reenactment of the continuum hypothesis. How many things lie between 0 and 1, how many things lie outside that range, and how do you compare the two.

As an engineer, I'm happy to say that if you make a random toss at the real numbers, the probability of landing in the interval 0 and 1 is zero, and that is that.
Proof? assign an integer to each interval, closed on one side and open on the other. Now you have reduced the problem to picking any one integer from the set of all integers, so the probability is 1/(countable infinity), which is zero.
However, the mathematicians are very interested in how to measure the sizes of different infinities. Check out the discussion in one of my posts if you actually want more: Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning.

If you are trying to make the shot.. Lets assume that your throws will be 'normally distributed', so the probability will follow a bell curve, centered on your aiming point. Integrate the area under that curve (the probability density function) over the target area (in the number line example, you aim for 0.5, and you integrate the curve from 0 to 1), and you have the probability of a hit. The reason this doesn't bring up issues of infinity is that the tails of the bell curve converge to zero very quickly. So, you don't have to worry about the possibility of missing by a long way, even though it is an infinitesimal possibility (adding up an infinite number of infinitesimal things is why I said it was a calculus question).
posted by Chuckles at 12:38 AM on November 22, 2006

The reason this doesn't bring up issues of infinity is that the tails of the bell curve converge to zero very quickly. So, you don't have to worry about the possibility of missing by a long way, even though it is an infinitesimal possibility (adding up an infinite number of infinitesimal things is why I said it was a calculus question).

Ugh! I totally should have re-written that..

This problem looks so much different because the tails of the bell curve converge to zero quickly. Basically, you don't have to worry about missing by a long way, because the chance of that is infinitesimal. Of course it all depends on the shape of the distribution, if the probability density doesn't decrease quickly enough with distance, you have the first question.
posted by Chuckles at 1:03 AM on November 22, 2006

I'm amazed that there's still continuing debate over the whole infinity thing. We've sufficiently advanced mathematics to handle it. Depending on your philosophy of how you use it, it's the limit of what happens as you increasingly finely consider the possible trajectories - using an ever smaller pixel scale to represent reality if you like.

I think zaebiz is totally nuts for suggesting Monte Carlo simulation as the only thing you can trust. I'm sure he'd agree that you want maxiter as large as possible though, and the techniques for handling continuous distributions are really just ways of expressing that the answer you get converges to something when your 'pixellisation' and number of iterations rise ever higher. The very notion of probability is entrenched in the infinite, being pretty much defined as the ratio as the number of attempts goes towards infinity.

I would like to retract something I said earlier - that there's no evidence of the distribution of trajectories not being continuous. There is one very nice experiment that showed that trajectories were not continuous in one very nice experiment. As explained here there was a rather nifty experiment involving throwing neutrons around inside a confined space. Thanks to quantum mechanics, because they were in a confined space they could only have a limited set of energies, and this was experimentally confirmed as there being no neutrons at all taking certain trajectories that should have been available to them under classical mechanics.

Now it's not clear to me if we can be sure this scales up to rocks and trashcans, and there are indeed a whole bunch of questions I'd want answering before being confident of that, but it's a very impressive hint towards the set of trajectories being non-infinite.
posted by edd at 1:20 AM on November 22, 2006

Allow me to repeat myself:

As an aside: if you don't know what an infinite cardinal or ordinal is, please refrain from writing things about infinity. All you're going to do is confuse people with your rambling nonsense. The only people qualified to answer this question are those who have some rigorous understanding of probability/measure theory.

Chuckles, please stop confusing the issue. This problem is not about the continuum hypothesis at all. It is a basic question of probability. Once you specify the distribution function for the initial velocity (by which I mean speed and direction), you just integrate over the set of initial velocities that actually get the stone into the can, and that is the probability. Or I suppose you could simplify it by just specifying the distribution function for where the stone lands.

And to be clear, the set of initial conditions which allow the stone to fall in the can is infinite. In fact, uncountably infinite. Proof: choose an initial angle A for which the shot can be made. Let v_min be the smallest velocity which will get the stone into the can when throwing with initial angle A, and let v_max be the largest such velocity. The can has a positive size, so v_min < v_max. Then any velocity between v_min and v_max will also get the stone into the can. Since the set of real numbers between v_min and v_max is uncountable, there are uncountably many initial conditions which will work.
posted by number9dream at 1:30 AM on November 22, 2006

edd: Yes, you're correct. We have made a simplifying assumption that velocities are continuous. But I think it's pretty sound at the non-quantum scale of rocks and trash cans.
posted by number9dream at 1:32 AM on November 22, 2006

number9dream: I think we're in complete agreement. Regardless of how nature really works, anyone actually trying to get an answer without doing things using continuous distributions is doing it the hard way.
posted by edd at 1:36 AM on November 22, 2006

edd: "I think zaebiz is totally nuts for suggesting Monte Carlo simulation as the only thing you can trust. I'm sure he'd agree that you want maxiter as large as possible though"

See the thing that non-computational mathematicians forget is that even if you wanted to work out the probability using mathe-magical notions such as infinity, limits and infinitesimals, it is highly unlikely that the analytical formula you come up with will not involve a calculation that requires a computational approximation to an infinite series to calculate in any event.

Real number analysis is just a shorthand signposting of reality invented in the days when mathematicians only had paper to work with.
posted by zaebiz at 1:51 AM on November 22, 2006

Yet if both are infinite, are they not the same probability? Are there not, in fact, infinite ways to achieve the former? Or is there something else I'm not getting?

What 777 is not getting is that infinity isn't just a number. Cardinality explores the fact that there are different infinities, and the continuum hypothesis shows that mathematics has a hard time comparing the infinity bounded by a finite interval, which is hitting the target, to the unbounded infinity of all other possibilities.

Granted, that doesn't help in calculating the probability.. That part of my answer wasn't intended to.
posted by Chuckles at 8:38 AM on November 22, 2006