Two aspects of randomness

Question

Consider a random sequence of integers

1, 4, 3, 8, 2, 5, 3, 8 ...

The only sufficient condition for the sequence to be random is its unpredictability ie. probability of any number coming next must be equal to $\frac{1}{10}$.
Now consider that we are getting only numbers less than 5 in the sequence, it then implies that for the sequence to be random the probability of getting numbers greater than 5 is now more, this does not follow the randomness criteria as numbers are now in some form more predictable.

Do the two aspects of randomness contradict with each other?
Or am I wrong somewhere in this deductive thinking?

Answer 1

Welcome to MSE,

it then implies that for the sequence to be random the probability of getting numbers greater than 5 is now more.

This is not true. If every choice of digits is independent, there is no change in the probabilities for the next digit of the sequence.

You can take a look at this question, which is somehow close to yours.

Does the probability change if you know previous results?

If that's not what you are asking please provide us with more details.

Answer 2

You confuse probability of the next drawing, which is always $\dfrac1{10}$ (provided the distribution is uniform and the samples are independent), and the probability of certain sequences.

For instance, the probabilities of drawing $1111111111$ or $1234567890$ or $5369574581$ are all $10^{-10}$, i.e. they are extremely unlikely events.

Now you can consider other events, such as all digits appearing exactly once among $10$ drawings; this is $10!\cdot 10^{-10}=0.00036288$ because you can permute $1234567890$ in $10!$ ways.

Or having as many digits below $6$ than above $5$ among $10$ drawings, which is $\displaystyle\binom{10}5\cdot2^{-10}=0.24609375$.

And so on.

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Randomness is indeed unpredictability. It does not require the outcomes to be equiprobable, nor independent of each other.

Answer 3

The only sufficient condition for the sequence to be random is its unpredictability

"Randomness" in the context of probability models^†, unlike in ordinary speech, has in a sense a contrary meaning to "haphazard and unpredictable".

After all, a stochastic process (which describes a random phenomenon) evolving in space or time does have an underlying structure, and so can be used for forecasting: while individual trials are unpredictable, by the law of large numbers, the process's underlying pattern (the relative frequencies of different outcomes) emerges over numerous trials.

probability of any number coming next must be equal to $\frac1{10}.$

This process, as in classical probability, involves equally-likely outcomes: as such, the next digit is a uniformly—not haphazardly—distributed random variable.

consider that we are getting only numbers less than 5 in the sequence, it then implies that for the sequence to be random the probability of getting numbers greater than 5 is now more

This is the Gambler's Fallacy—based on the false implicit assumption that small samples are representative of the larger population—a misapplication of the Law of Large Numbers.

^{† To what extent probability models correspond to the actual workings of reality or actual randomness—and to what degree this question is simplistic—is a different, philosophical discussion}