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Probability Spaces with Infinitely Many Outcomes

To this point, we have focused entirely on probability spaces (S,P) with S a finite set. More generally, probability spaces are defined where S is an infinite set.

Probability Spaces with Infinitely Many Outcomes

To this point, we have focused entirely on probability spaces \((S,P)\) with \(S\) a finite set. More generally, probability spaces are defined where \(S\) is an infinite set. When \(S\) is countably infinite, we can still define \(P\) on the members of \(S\), and now \(\sum_{x\in S} P(x)\) is an infinite sum which converges absolutely (since all terms are non-negative) to \(1\). When \(S\) is uncountable, \(P\) is not defined on \(S\). Instead, the probability function is defined on a family of subsets of \(S\). Given our emphasis on finite sets and combinatorics, we will discuss the first case briefly and refer students to texts that focus on general concepts from probability and statistics for the second.

Example

Consider the following game. Nancy rolls a single die. She wins if she rolls a six. If she rolls any other number, she then rolls again and again until the first time that one of the following two situations occurs: (1) she rolls a six, which now this results in a loss or (2) she rolls the same number as she got on her first roll, which results in a win. As an example, here are some sequences of rolls that this game might take:

\((4,2,3,5,1,1,1,4)\). Nancy wins!
\((6)\). Nancy wins!
\((5,2, 3,2,1,6)\). Nancy loses. Ouch.

So what is the probability that Nancy will win this game?

Nancy can win with a six on the first roll. That has probability \(1/6\). Then she might win on round\(n\) where \(n\ge2\). To accomplish this, she has a \(5/6\) chance of rolling a number other than six on the first roll; a \(4/6\) chance of rolling something that avoids a win/loss decision on each of the rolls, \(2\) through \(n-1\) and then a \(1/6\) chance of rolling the matching number on round\(n\). So the probability of a win is given by: \[\begin{aligned}\end{aligned}\]

Example

You might think that something slightly more general is lurking in the background of the preceding exampleand it is. Suppose we have two disjoint events \(A\) and \(B\) in a probability space \((S,P)\) and that \(P(A)+P(B)\lt 1\). Then suppose we make repeated samples from this space with each sample independent of all previous ones. Call it a win if event \(A\) holds and a loss if event \(B\) holds. Otherwise, it's a tie and we sample again. Now the probability of a win is: \[\begin{aligned}\end{aligned}\]

Symbols used here

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

P(A),\ P(A \mid B)

probability, conditional probability
Chance of A; chance of A given that B happened.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\bar{x},\ \mu

sample mean, population mean
Average of the data; average of the whole population.

\sigma,\ s,\ \sigma^2

standard deviation, sample s.d., variance
Typical distance from the mean; its square.

E[X],\ \operatorname{Var}(X)

expected value, variance
Probability-weighted average of X; its spread.

N(\mu, \sigma^2),\ z

normal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.

\mu(A),\ \sigma\text{-algebra}

measure of A
Size of a set; the family of sets that can be measured.

Questions people ask

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.

Cubalah sendiri

Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

Lebih dalam Probability Theory

Sample spaces and the axioms Random variables and expectation The common distributions The law of large numbers and the central limit theorem