# Why is the probability of a continuous variable taking a particular value zero? Explain only logically

This question isn't the same as this

I don't want a mathematical proof or something of the sort. I want a verbal explanation that intuitively will convince me why this is true.

The way I see it, when you plot a continuous function like say the weight of all the people in the world. This will be continuous, right? Now for any given weight w, there will have to be some number of people who have that particular weight right? So why will the probability of someone having that weight w be zero?

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This will be continuous, right? No, this will not be, and this cannot be since all the people in the world is a finite set. – Did Oct 3 '12 at 6:47
I'd say any function on a finite set is continuous – when equipped with discrete topology. – k.stm Oct 3 '12 at 6:48
"Why is the probability $\ldots$ zero?" vs. "I don't want a mathematical proof or something of the sort.": You can not eat the cake and have it. – Christian Blatter Oct 3 '12 at 7:48
You shouldn't expect intuitive explanations where the reals are involved. They are unlike anything in our day-to-day experience, so our intuition serves us poorly. – AakashM Oct 3 '12 at 9:07
@K.Stm. Quite true and quite off-the-point. – Did Oct 6 '12 at 15:12

The distribution of the weights of people in the world is of course discrete: there is only a finite number of distinct weights associated with the finite number of people that exist. However, you can certainly consider a continuous probability distribution, like the bell curve, which approximates the probability that a randomly chosen person has a given weight. In this idealized world, a random person's weight may be an arbitrary real number, and this number can be known to arbitrary accuracy.

Now there is some finite probability that our random weight lies between 70 kg and 80 kg. Then the probability that it lies between 70 and 71 kg is roughly a tenth of that, because we're considering only one-tenth of the interval, right? And the probability between 70.0 and 70.1 kg is a tenth further, and between 70.00 and 70.01 kg a tenth still. What do you suppose that leaves the probability that the weight is exactly 70.000... kg?

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In fact, if you consider all the weights of all the people in the world in the past year, it is a continuous distribution if weights changes continuously, and you might consider their weights right now as a sample from this distribution. – Henry Oct 3 '12 at 7:12
But there have got to be some people whose weights are exactly 70 right? It just logically doesn't make sense to be. The math is fine but intuitively I find it difficult to digest. – Programming Noob Oct 3 '12 at 13:53
@ProgrammingNoob: Do you think there also have to be some people whose weights are exactly 70.010000..., or 71.666666..., or 73.141592...? – Rahul Oct 3 '12 at 21:56

This is because you have a finite amount of people to check this. The requirement for a distribution on a finite set to be continuous isn't a requirement at all, since in any function on a finite set (equipped with the discrete topology) is continuous. But this wouldn't give you a continuous distribution in the sense of your statement. This would require the range to be some sort of continuum.

Rather consider saying "now" in a particular moment in time, measured 100% accurately (if that was possible). Or maybe throwing a dart at a wall, hitting a particular spot on it.

Hope that helps.

Actually, it depends if you distribute weight over the people in the world or people of the world over possible weights they have. If you do the latter, the answer from Rahul would be the right one.

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Where is Rahul's answer? He explained that the probability of someone to have exactly weight 70.000...kg is $0$. – k.stm Oct 3 '12 at 6:59
I deleted and undeleted it. – Rahul Oct 3 '12 at 7:08

In intuitive terms: Let's say I ask you to throw a dart at a dartboard. To keep things simple, assume the face of the dartboard has an area of 1 square meter, and that you are equally likely to hit any spot on the dartboard. What is the probability that the dart will land at a particular point on a dartboard?

We can approximate this by discretizing the dartboard and thinking of it as a bunch of tiny squares that are 1 millimeter on each side. In that case, the probability of landing in a specific square is 1 in a million. But this is an approximation, and we're not really happy with it, so perhaps we can divide the dartboard into "points" that are 1 micrometer on each side. Now that we have a finer granularity, the chances of hitting a specific point are 1 in a trillion.

Of course, this is still an approximation, and we want to be exact, so let's look at individual molecules on the dartboard's surface. What is the probability of hitting a specific molecule? It'll be a tiny number -- maybe something like 10^-30. In the pursuit of accuracy, we can go beyond molecules to atoms or even to quarks. As we get closer and closer to defining a point as being infinitesimally small, the probability of hitting that point gets infinitesimally close to 0.

In a continuous distribution, a "point" is the same as a "point" the dartboard -- it's dimensionless. You can get non-zero probabilities if you approximate the point with some interval, but as the interval shrinks your probability will approach 0.

In math terms: Probability of an event = (# of outcomes that lead to that event) / (total # of outcomes)

In a continuous distribution, there's an infinite number of outcomes, so the probability of a specific outcome is 1/infinity, which is 0.

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