I take a piece of paper, break it into two pieces. Is it true that mathematically speaking, the chance of one piece being exactly 5cm is 0? (In fact in that case the chance of any value coming up is 0, which is a horrible paradox...)
2006-07-11 02:20:59 · 31 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Yes, that's true. Let's say you're going to measure where the paper breaks to the nearest cm, and the paper is 10 cm long. That means the probability of breaking at 5 cm is 1/10 because you have defined 10 possibilities. Now let's say you're going to measure to the nearest mm. Now the probability of breaking at any particular spot is 1/100. You can keep shrinking your unit of measurement to get a more accurate anser until you have an impossiblly small unit, and there are an infinite number of units in the length of the paper. The probability of breaking at one exact point is now 1/infinity, which equals zero by definition. This kind of probability is called a continuous probability. Other continuous probabilities use these kinds of units of measure: volume, area, temperature, etc. In a continuous probability, most people figure the probability of getting a value within a certain range, such as the probability of the paper breaking somewhere between 4.9 and 5.1 cm.
2006-07-11 02:29:06 · answer #1 · answered by Anonymous · 0⤊ 1⤋
Not quite.
What you are imagining is that the length of the piece of paper after the tear will be random in a continuous distribution. That is, its length may be any real number within a particular range. (There's more to the definition of continuous distributions than that, but you get my meaning.) If you accept that assumption, then yes, you're right - any _specific_ number has an infinitesimal (effectively zero) chance of occurring.
However, there are three reasons why that doesn't apply to your piece of paper.
First, when you break a real object into two pieces, the break can only occur at specific positions - at least, at breaks between atoms. Yes, there are probably over 10^22 possible positions between atoms, but there's still only a finite number.
Second, the length isn't exactly well-defined. You've heard of Heisenberg's Uncertainty Principle, I assume - an implication of it is that the exact borders of the piece of paper are impossible to determine exactly.
Third, your measuring tool isn't perfect. If you're using a regular old ruler, you probably can't measure more precisely than (at best) 0.1mm, and even calipers won't improve that more than a few orders of magnitude. Therefore, an entire (very small) range of lengths will count as exactly 5cm.
To wrap it up: if you're in a math class, then yes, the probability is zero. But in the real world, it isn't.
P.S. The paradox you mention isn't actually a problem. While each specific number has a zero chance of happening, each range has an infinite number of numbers. Thus, the chance of getting in that range is zero times infinity, and therefore can be a number.
2006-07-11 02:35:37 · answer #2 · answered by peri_renna 3 · 0⤊ 0⤋
Not entirely true. The chances are so amazingly, incredibly slim that one may as well round to zero. It's a question of accuracy and probability.
It's improbable that one piece will be a given length because there are so many other lengths. And the scope of the size is important. One may say that the piece is 5 cm, but find under a microscope that it is, in fact, 5.0001 cm, thus making the odds of one piece being exactly 5 cm extremely unlikely. But, because 5 cm is a defined length, the odds cannot be zero, or there would be no point in having such a silly thing as a "centimeter".
2006-07-11 02:25:02 · answer #3 · answered by interested 2 · 0⤊ 0⤋
No. For a start you didn't say 5cm squared or anything like as such.
It obviously depends on the size of the original piece of paper.
If only the length or width or depth have to be 5cm then the probability goes up.
There is not enough information in the question to make an accurate answer!
2006-07-12 08:25:00 · answer #4 · answered by ridcully69 3 · 0⤊ 0⤋
This is not a probability question. If you are talking about size, then 5cm is not the dimension for an area. If length, and the original was also 5 cm, and torn along length, then it will definitely be 5 cm. Or definitely not be 5 cm. Probability is the science of finding the possibilities of favourable outcomes from the possible outcomes. But my daughter described it best, when I was teaching her saying,
The event either happens or not at all. So the probability is 50% in all cases. Or I remember a demonstration by Simon Singh who explained how probability could be manipulated
2006-07-11 02:29:49 · answer #5 · answered by mkaamsel 4 · 0⤊ 0⤋
The answer is indeed 0, as it would be for any other single value. The length of any piece is a continuous random variable and the probability that such a r.v. assumes any given value (rather than values in an interval) is necessarily 0...but probability of 0 does not mean impossible! In more technical terms it means that the relative size of the success space (its measure) is 0 compared with the possibility space. Hope this helps.
2006-07-11 05:02:50 · answer #6 · answered by Anonymous · 0⤊ 0⤋
The chance is not zero.
If you randomly tear a piece of paper in two, there is a chance that you can measure one of the pieces at 5 cm. If your measuring instrument is very poor (e.g. only to the nearest centimeter) then the chance is much greater than if your measuring instrument can measure to the nearest angstrom.
Regardless of the precision of your measuring instrument, there is some chance that you will measure one of the torn pieces at exactly 5 cm.
You cannot necessarily generalize that just because "x" can't happen, that nothing can happen. But there is no need for that discussion here, because certainly "x" can happen.
2006-07-11 02:36:59 · answer #7 · answered by Steve W 3 · 0⤊ 0⤋
Yes.
Think of this another way. Suppose you had a number line running from 0 to 10 cm, and the number line can represent as small of a number as you like, such as 0.000000000000000001 cm.
Suppose you had a pencil with a point so sharp that it could point to a number that small. If you tossed the pencil point at random to the number line, the chance of ithitting 5.00000000000000000 cm, and not a neighboring point, is zero.
The way to get around this is to ask what the probability is of being in an interval, of say between 4.9 and 5.1 is. That would be a non-zero number.
2006-07-11 02:28:34 · answer #8 · answered by fcas80 7 · 0⤊ 0⤋
Short answer, yes ... The probability density function p(x) for this process is a continuous function, representing essentially an infinite number of possibilities for lengths of that one piece (between 0 and the length of the original piece of paper) -- What might make more sense to talk about is the probability that the piece you tore off is between 4cm and 6cm in length. If you had some description of the continuous probability density function, you could integrate that function from 4 to 6 to evaluate the probability that your piece was of length between 4 and 6cm. Hope this helps!
2006-07-11 02:31:04 · answer #9 · answered by maverick 1 · 0⤊ 0⤋
It is zero if it is smaller than 5cm to start with. There is a probability of it being 5cm, but it is very small.
Like the cr@p about an infinite number of monkeys and an infinite numberof typewriters coming up with the complete works of Shakespeare. Yes, it would happen in theory, but monkeys would probably be extinct by then
2006-07-11 02:25:50 · answer #10 · answered by izzieere 5 · 0⤊ 0⤋