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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?

Mauro ALLEGRANZA
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Abhishek Choudhary
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3 Answers3

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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.

Gâteau-Gallois
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  • isn't it expected from a series of coin tosses half of them must be head and half must be tails,ie. if i toss coin 1000 times i expect 500 heads and 500 tails, if after 800 tossed i have 700 heads doesn't it mean that to maintain 1:1 ratio tails are more expected, atleast mathematically. – Abhishek Choudhary Nov 17 '18 at 07:04
  • Precisely not. The tosses are independent. Consider the case with 3 tosses. You have $3^2 = 9$ possible outcomes (if you take care of the order of the tosses). Each outcome is obtained with probability $\frac{1}{9}$. If you already had head twice, it means that you are either in the situation HHH or in the situation HHT. But both of those situations have the same probability to be obtained, hence you have the same probability to obtain H or T at the last toss. – Gâteau-Gallois Nov 17 '18 at 17:41
  • then why do we get 50:50 heads and tails in a large sequence of tosses – Abhishek Choudhary Nov 18 '18 at 06:24
  • You're mixing two different things. If you have no previous informations about the sequence, and toss a coin $10^9$ times, you expect the number of heads to be close to the number of tails. This is basically the Law of Large Numbers. On the other hand, what the law of Large numbers say is just that it's very unlikely (but not impossible) to have for instance $10^9$ heads and 0 tails. – Gâteau-Gallois Nov 18 '18 at 13:28
  • Now if you toss a coin 800 times and get 800 heads, you are just in one of those ``less likely'' scenarii. And you might end up in the scenario with 800-200 or with 1000-0. Both are actually equally likely if you already know that the first 800 tosses have given heads. And they are both part of those unlikely scenarii in the law of large numbers for which the ratio is not close to 1:1. – Gâteau-Gallois Nov 18 '18 at 13:32
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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.

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

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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 a meaning contrary to ‘unpredictable’ and to ‘arbitrary’††.

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: the next digit is not arbitrary but a uniformly 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

†† Arbitrary vs. Random

ryang
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