mathematics

Impossibility of a gambling system

Impossibility of a gambling system is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Impossibility of a gambling system rather than just read about it. In short: The principle of the impossibility of a gambling system is a concept in probability. It states that in a random sequence, the methodical selection of subsequences does not change the probability of specific elements.

Key takeaways

Impossibility of a gambling system belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Impossibility of a gambling system to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Impossibility of a gambling system from memory before moving on to harder problems.

Reference excerpt

The principle of the impossibility of a gambling system is a concept in probability. It states that in a random sequence, the methodical selection of subsequences does not change the probability of specific elements. The first mathematical demonstration is attributed to Richard von Mises (who used the term collective rather than sequence). The principle states that no method for forming a subsequence of a random sequence (the gambling system) improves the odds for a specific event. For instance, a sequence of fair coin tosses produces equal and independent 50/50 chances for heads and tails. A simple system of betting on heads every 3rd, 7th, or 21st toss, etc., does not change the odds of winning in the long run. As a mathematical consequence of computability theory, more complicated betting strategies (such as a martingale) also cannot alter the odds in the long run. Von Mises' mathematical demonstration defines an infinite sequence of zeros and ones as a random sequence if it is not biased by having the frequency stability property. With this property, the frequency of zeroes in the sequence stabilizes at 1/2, and every possible subsequence selected by any systematic method is likewise not biased. The subsequence selection criterion is important, because although the sequence 0101010101... is not biased, selecting the odd positions results in 000000... which is not random. Von Mises did not fully define what constituted a "proper" selection rule for subsequences, but in 1940 Alonzo Church defined it as any recursive function which having read the first N elements of the sequence decides if it wants to select element number N+1. Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability. In the mid-1960s, A. N. Kolmogorov and D. W. Loveland independently proposed a more permissive selection rule. In their view Church's recursive function definition was too restrictive in that it read the elements in order. Instead they proposed a rule based on a partially computable process which having read any N elements of the sequence, decides if it wants to select another element which has not been read yet. The principle influenced modern concepts in randomness, e.g. the work by A. N. Kolmogorov in considering a finite sequence random (with respect to a class of computing systems) if any program that can generate the sequence is at least as long as the sequence itself.

See also Gambler's ruin History of randomness No free lunch theorem

References

Illustrations

Worked examples

Example 1 — a first encounter with Impossibility of a gambling system

Start with the simplest possible case. Write down what Impossibility of a gambling system claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Impossibility of a gambling system before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Impossibility of a gambling system ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Impossibility of a gambling system

In research: Impossibility of a gambling system appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Impossibility of a gambling system in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Impossibility of a gambling system is common in secondary-school and first-year university syllabi. It links to neighbouring topics Probability theory, so understanding it makes those chapters shorter.
In everyday life: Look for Impossibility of a gambling system outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Impossibility of a gambling system in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Impossibility of a gambling system means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Impossibility of a gambling system out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Impossibility of a gambling system in simple terms?

Why does Impossibility of a gambling system matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Impossibility of a gambling system?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Impossibility of a gambling system.