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An Introduction to Probability
We continue with an informal discussion intended to motivate the more structured development that will follow. Consider the spinner shown in . Suppose we give it a good thwack so that the arrow goes round and round.
An Introduction to Probability
We continue with an informal discussion intended to motivate the more structured development that will follow. Consider the spinner Suppose we give it a good thwack so that the arrow goes round and round. We then record the number of the region in which the pointer comes to rest. Then observers, none of whom have studied combinatorics, might make the following comments:
The odds of landing in region\(1\) are the same as those for landing in region\(3\).
You are twice as likely to land in region\(2\) as in region\(4\).
When you land in an odd numbered region, then 60% of the time, it will be in region\(5\).
We will now develop a more formal framework that will enable us to make such discussions far more precise. We will also see whether Alice is being entirely fair to Bob in her proposed game to one hundred.
We begin by defining a probability space as a pair \((S,P)\) where \(S\) is a finite set and \(P\) is a function that whose domain is the family of all subsets of \(S\) and whose range is the set \([0,1]\) of all real numbers which are non-negative and at most one. Furthermore, the following two key properties must be satisfied:
\(P(\emptyset)=0\) and \(P(S)=1\).
If \(A\) and \(B\) are subsets of \(S\), and \(A\cap B=\emptyset\), then \(P(A\cup B)= P(A)+P(B)\).
When \((S,P)\) is a probability space, the function \(P\) is called a probability measure, the subsets of \(S\) are called events, and when \(E\subseteq S\), the quantity \(P(E)\) is referred to as the probability of the event \(E\).
Note that we can consider \(P\) to be extended to a mapping from \(S\) to \([0,1]\) by setting \(P(x)=P(\{x\})\) for each element \(x\in S\). We call the elements of \(S\) outcomes (some people prefer to say the elements are elementary outcomes) and the quantity \(P(x)\) is called the probability of \(x\). It is important to realize that if you know \(P(x)\) for each \(x\in S\), then you can calculate \(P(E)\) for any event \(E\), since (by the second property), \(P(E)=\sum_{x\in X}P(x)\).
Example
For the spinner, we can take \(S=\{1,2,3,4,5\}\), with \(P(1)=P(3)=P(4)=1/8\), \(P(2)=2/8=1/4\) and \(P(5)=3/8\). So \(P(\{2,3\})=1/8+2/8=3/8\).
Example
Let \(S\) be a finite, nonempty set and let \(n=|S|\). For each \(E\subseteq S\), set \(P(E)=|E|/n\). In particular, \(P(x)=1/n\) for each element \(x\in S\). In this trivial example, all outcomes are equally likely.
Condensed: the full section is in Keller & Trotter, Applied Combinatorics.
Tani ti. Asnjë kalkulator nuk e zgjidh këtë, por pjesët e saj janë të llogaritura. Provo një më poshtë, ose shkruaj tënde.
Një llogari e lirë shtohet shënime në çdo mësim, një regjistrim të asaj që ju keni përfunduar, problemet tuaja të zgjidhura në një vend, dhe një mësues që ju mund të pyesni rreth kësaj faqeje. Matematika vetë është e hapur për të gjithë, të regjistruar apo jo.
Regjistrohu HyrSimbolet e përdorura këtu
Prek çdo simbol për përkufizimin e plotë, një fotografi dhe se çfarë do të thotë çdo shkronjë në të.
Pyetja që bëjnë njerëzit
What is the difference between probability and statistics?
Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.
What does the law of large numbers promise?
That the average of many independent samples converges to the expected value. It says nothing about any single trial.
Pjesa e kësaj faqeje është adaptuar nga Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). E përmbledhur dhe ri-shkruar këtu; gabimet janë tona.
Më shumë në Probability Theory
Sample spaces and the axiomsRandom variables and expectationThe common distributionsThe law of large numbers and the central limit theorem