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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 shown in . 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:

  1. The odds of landing in region\(1\) are the same as those for landing in region\(3\).

  2. You are twice as likely to land in region\(2\) as in region\(4\).

  3. 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:

  1. \(P(\emptyset)=0\) and \(P(S)=1\).

  2. 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.

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.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
P(A),\ P(A \mid B)
probability, conditional probability
Chance of A; chance of A given that B happened.
n!
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).
\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.

Try your own

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

More in Probability Theory

Sample spaces and the axiomsRandom variables and expectationThe common distributionsThe law of large numbers and the central limit theorem