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If I draw a number uniformly between zero and one, what is the probability that they are equal? Mathematically, it should be zero but I don't recall why? Can somebody please help in explaining why the probability should be zero?

For example, there is still a chance, that when generating six random numbers they will be all 1 or all 0? So if the probability is zero then why in digital communications do we assume that the symbols are $iid$ and that if the symbols are 0 and 1, 50% of them will be 1 and the rest will be 0 i.e., why do we assume equi-probable symbol probabilities when theoretically they cannot be equal? In Matlab, I tried my best to generate a large sequence of 0/1 but never did I get equi-probable symbol probabilities.

Am I misunderstanding some thing here?

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If I draw a number uniformly between zero and one, what is the probability that they are equal? Mathematically, it should be zero but I don't recall why? Can somebody please help in explaining why the probability should be zero?

Avoiding formal definitions of (Lebesgue) probability measure, an informal way is thinking the probability at a point of a uniformly distributed random variable is $\frac{1}{\textrm{number of points}}$. Then the number of points of real values between zero and one is infinity, thus the probability must be $\frac{1}{\infty} = 0$.

For example, there is still a chance, that when generating six random numbers they will be all 1 or all 0? So if the probability is zero then why in digital communications do we assume that the symbols are $iid$ and that if the symbols are 0 and 1, 50% of them will be 1 and the rest will be 0 i.e., why do we assume equi-probable symbol probabilities when theoretically they cannot be equal? In Matlab, I tried my best to generate a large sequence of 0/1 but never did I get equi-probable symbol probabilities.

The previous arguments change if we divide the interval $[0,1]$ to several sub-interval of width $\Delta$. Thus the probability falling on each sub-interval is $\frac{1}{1/\Delta}=\Delta \neq 0$.

In your example, $\Delta = 0.5$ yields two sub-intervals and each has probability $0.5$. That's where the equiprobable symbol assumption holds.

Experimentally, e.g. with Matlab, you (almost surely) never get exactly the probability $\Delta$ because the number of tests is always finite.

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