What exactly is randomness?

Question

I recently read somewhere that mathematicians are unable to define or prove randomness. and although some sequences of numbers are categorized as random, these are only educated guesses...Please enlighten.

Answer 1

Randomness is really just what you think it is. A sequence of 32-bit integers is random if the next number in the sequence can be any 32-bit integer with equal probability, and the sequence is totally unpredictable.

Computer-based random number generators are useful for a variety of tasks. Statistical simulations is a primary example. A good random number generator produces a sequence that is 'good enough' for the application. If you can detect a pattern in its output, it is not a good random number generator. For example, it generates only odd integers, or the sequence repeats every 17,533 numbers. One which repeats every 3,447,655,328 numbers is better but is still not random. Proving a sequence is random requires checking it forever to make sure it does not repeat. Obviously that is not possible in the general case.

Answer 2

Random numbers are those which cannot be predicted by statistical means -- any mathematical analysis applied to a set of such numbers results in a distribution similar to the original distribution. This implies that one needs specific application-dependent tests to determine if the numbers are sufficiently random for the application at hand, sucn as Monte Carlo anaysis or simulation of nuclear processes. Radioactive decay is an example of a natural random process; the time of decay of a particular atom is totally unpredicable, even though the half-life of a sample can be measured with great precision. If you need a random number for a computer application, measuring the time between user keystrokes to the nearest microsecond, and taking a few bottom digits of this, would give a number which is random enough for almost any application, such as running an electronic slot gambling machine.

Answer 3

In order to prove a sequence of numbers is random, you would need to show that there are no patterns in the numbers. Perhaps there is a pattern, but you just haven't found it. You can't prove that you have tested for all possible patterns, so you can't prove the sequence is random.

Answer 4

ran‧dom  /ˈrændəm/ Pronunciation Key - Show Spelled Pronunciation[ran-duhm] Pronunciation Key - Show IPA Pronunciation

–adjective 1. proceeding, made, or occurring without definite aim, reason, or pattern: the random selection of numbers.
2. Statistics. of or characterizing a process of selection in which each item of a set has an equal probability of being chosen.
3. Building Trades. a. (of building materials) lacking uniformity of dimensions: random shingles.
b. (of ashlar) laid without continuous courses.
c. constructed or applied without regularity: random bond.

–noun 4. Chiefly British. bank3 (def. 7b).
–adverb 5. Building Trades. without uniformity: random-sized slates.
—Idiom6. at random, without definite aim, purpose, method, or adherence to a prior arrangement; in a haphazard way: Contestants were chosen at random from the studio audience.