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Two real numbers $a, b \in [0, k]$.
What is the probability for $a = b$ ?


The probability is uniform, so:

$$ P(a=b) = \lim_{n \to ∞} \frac n {n^2} = \frac 1 n = 0 $$

But $a$ can equal $b$ because they are in the same domain.
So why is $P(a=b) = 0$ ?

Rasmus
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    Saying $P(E) = 0$ is not the same as saying that the event $E$ is impossible. – Travis Willse Oct 29 '15 at 22:16
  • Why is that? (this string is just to fill 15 characters) – Rasmus Oct 29 '15 at 22:21
  • For any $c \in [a, b]$, $a < b$, the probability that a uniformly randomly selected number in $[a, b]$ is $c$ is zero. If this meant that the event that the selected number is $c$ is impossible, then since $c$ is arbitrary... – Travis Willse Oct 29 '15 at 22:25
  • See this: http://math.stackexchange.com/questions/41107/zero-probability-and-impossibility. – Eric Brooks Oct 29 '15 at 22:36

1 Answers1

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The sample space here is the product space $[0,k]\times[0,k]$ with the uniform probability on the rectangle $[0,k]\times [0,k]$, which gives to every (measurable) subset it's area. The event $a=b$ is simply the diagonal of the rectangle, which has area $=$ zero, hence probability $=$ zero.

uniquesolution
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  • Probability is relative frequency after a long time.Does zero probability mean that the event will not repeat? No matter how many times one does the attempt? – Rasmus Oct 29 '15 at 22:38
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    No, it does not mean that the event will never occur. In any physical experiment, it could happen that on your very first choice of a point in the rectangle, you stumble across a point that belongs to the diagonal. Events that have probability zero can happen. To prove that an event has probability zero by relative frequency when you are only allowed finitely many experiments is mathematically impossible. You will get a very small relative frequency, though. – uniquesolution Oct 29 '15 at 22:45