Consider the adhering to game: I write down a random genuine number between 0 and also 1, and also ask you come guess it. What’s the probability the you guess that correctly? The answer is zero. You could wonder: “But it’s possible for me to guess the correct answer! That method that the probability needs to be an ext than zero!” and you would certainly be justified in wondering, however you’d be wrong. It’s true that events that space impossible have zero probability, however the converse is not true in general. In the rest of this post, we present why the answer above was in reality zero, and why this doesn’t have to do irreparable damage to your current worldview.

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Let’s start by reflecting that the probability the you guess mine number appropriate is zero. Allow

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be the probability in question. The idea is to present that
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for any kind of positive real number
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. We know that
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, and if it’s smaller than any positive number, then it needs to be zero! The debate is together follows. Let’s contact the number i randomly picked
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. Imagine the the expression
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" class="latex" /> is painted white. Pick any type of positive real number
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. Then there is one sub-interval of length
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in ~ the interval
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" class="latex" /> include
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. Imagine the this sub-interval is painted black, so now we have actually a black color strip of size
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top top the initial white strip, and the number I made decision was in the black color strip. What’s the probability the your guess floor on the black strip? It needs to be
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, since that’s the relationship of the white piece that is covered. However in bespeak for her guess to same my number
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, it has to land in the black color strip, so your probability
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that guessing
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can’t be bigger than the probability of guessing a number ~ above the black strip! therefore
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.

You must now be encouraged that this event indeed has actually zero probability that happening, yet it’s still true. This phenomenon is due to the fact that of the complying with geometric fact: it’s possible to have a non-empty collection with zero “volume”.

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 The ax “volume” relies on the context; in the instance of the allude on the interval, “volume” is length. The probability of an event measured top top the interval

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" class="latex" /> is same to that length, and also a single point top top the interval has zero length, yet it’s quiet a non-empty subset the the interval! Probability is usually a measure up of “volume” where the entire space has “volume” same to 1. By specifying probability in this way, we have the right to prove every kinds of practiced facts using something dubbed measure theory.

To recap, friend should have learned the following from this post:

The probability that randomly choosing a particular number in the expression
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" class="latex" /> is equal to zeroEvents that have zero probability are still possible