The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times. ****

[!def] Law of Large Numbers As more observations are collected, the proportion \(p_n\) of occurrences with a particular outcome converges to the probability \(p\) of that income.

Two outcomes are called disjoint or mutually exclusive if they cannot both happen. For example, you cannot roll both a 1 and 2 on a die at the same time. We compute the probability that one of these outcomes will occur by adding their separate probabilities:

\(P(1 \text{ or }2) = P(1) + P(2) = 1/6 + 1/6 = 1/3\)

[!def] Addition Rule of Disjoint Outcomes If \(A_1\)and \(A_2\)represent two disjoint outcomes, then the probability of one of them occurring is given by \(P(A_1 \text{ or } A_2) = P(A_1) + P(A_2)\).

Sets of outcomes are often called events. For a die roll, we might have:

A = {1, 2} - B = {4, 6} - C = {5} - D = {2, 3}

A and B are disjoint events, since they do not share any outcomes. The Addition Rule applies to both outcomes and events.

Venn Diagrams¶

This Venn diagram represents all diamond cards on the left side, and all face cards on the right side.

Since these events are not disjoint, the Addition Rule does not work. Instead, we can use the Venn diagram.

Add the probabilities of the two events: \(P(A) + P(B)=P(\text{diamond})+P(\text{face card}) = 13/52 +12/52\)
The three cards that are in both events were counted twice, so we must correct: \(P(A)+P(B)=13/52+12/52-3/52=22/52 = 11/26\)

This is known as the general addition rule.

[!def] General Addition Rule If \(A\) and \(B\) are any two events, disjoint or not, then the probability that at least one of them will occur is \(P(A \text{ or } B)=P(A)+P(B)-P(A \text{ and } B)\), where \(P(A \text{ and }B)\) is the probability that both events occur.

Probability Distributions¶

A probability distribution is a table of all disjoint outcomes and their associated probabilities. It must follow these rules:

Outcomes listed must be disjoint
Each probability must be between 0 to 1
The probabilities must total 1

They can also be represented in bar plots.

Complement of an Event¶

The set of values that can be produced by rolling a traditional die is {1, 2, 3, 4, 5, 6}. This set is called the sample space (\(S\)) for rolling a die.

Let \(D\) be some subset of the sample space. The complement of \(D\) is all the outcomes in the sample space that are not \(D\), and is denoted by \(D^c\).

Independence¶

If two results are not related, i.e. knowing the outcome of one provides no useful information on the outcome of the other, they are independent.

Rolling two dice is a basic independent process.

[!def] Multiplication Rule for Independent Processes If \(A\) and \(B\) represent independent events, then the probability that both \(A\) and \(B\) occur can be calculated as the product of their separate probabilities: \(P(A \wedge B) = P(A) \cdot P(B)\) If there are \(k\) events \(A_1,...,A_k\) from \(k\) independent processes, the probability they all occur is \(P(A_1\wedge...\wedge A_k)=P(A_1)\cdot...\cdot P(A_k)\)