Probability Rules and Random Processes: Study Notes

Probability Rules

Objectives

• Understand random processes and the Law of Large Numbers

• Apply rules of probability

• Compute and interpret probabilities using the empirical method

• Compute and interpret probabilities using the classical method

• Recognize and interpret subjective probabilities

Random Processes and the Law of Large Numbers

Introduction to Random Processes

A random process is a scenario where the outcome of any particular trial of an experiment is unknown, but the proportion (or relative frequency) of a particular outcome is observed as the number of trials increases.

• Probability measures the likelihood of a random phenomenon or chance behavior occurring.

• Probability experiments yield random short-term results or outcomes, yet reveal long-term predictability.

• The Law of Large Numbers states that as the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of that outcome.

Simulation is a technique used to recreate a random event, often to measure how often a goal is observed.

Example: Rolling a Die

• Simulate rolling a six-sided die multiple times and record the proportion of times a specific outcome (e.g., rolling a 'four') occurs.

• As the number of rolls increases, the relative frequency of rolling a 'four' approaches the theoretical probability ($\frac{1}{6}$).

Example: Law of Large Numbers in Daily Life

• Track the number of days a traffic light is red during your commute over many days.

• Graph the cumulative proportion of red lights against the number of days.

• As the number of days increases, the proportion stabilizes, illustrating the Law of Large Numbers.

Law of Large Numbers vs. Law of Averages

• The Law of Large Numbers is often misinterpreted as the "Law of Averages," which is incorrect.

• Each trial is independent; previous outcomes do not affect future probabilities.

Probability Experiments and Sample Space

Definitions

• An experiment is any process that can be repeated in which the results are uncertain.

• The sample space ($S$) is the collection of all possible outcomes.

• An event is any collection of outcomes from a probability experiment.

Example: Rock, Paper, Scissors

• Sample space: {Rock, Paper, Scissors}

• Event $E$: Choosing one Rock

Rules of Probability

Basic Rules

• The probability of any event $E$, $P(E)$, must be $0 \leq P(E) \leq 1$

• The sum of the probabilities of all outcomes must equal 1: $P(e_1) + P(e_2) + \ldots + P(e_n) = 1$

Probability Model

A probability model lists all possible outcomes and each outcome's probability. It must satisfy the rules above.

Color	Probability
Brown	0.12
Yellow	0.16
Red	0.12
Orange	0.23
Green	0.15

Classifications

• If an event is impossible, $P(E) = 0$

• If an event is a certainty, $P(E) = 1$

• The closer $P(E)$ is to 1, the more likely the event will occur

• An unusual event is one with a low probability of occurring (commonly, $P(E) < 0.05$)

Compute and Interpret Probabilities Using the Empirical Method

Empirical Approach

The probability of an event $E$ is approximated by the number of times event $E$ is observed divided by the number of repetitions of the experiment:

$P(E) \approx \frac{\text{relative frequency of } E}{\text{number of trials of experiment}}$

Example: Build a Probability Model from a Random Process

Outcome	Frequency
Side with no dot	12344
Razorback	767
Trotter	365
Snouter	1377
Leaning Jowler	32

Use the frequencies to estimate probabilities for each outcome.

Compute and Interpret Probabilities Using the Classical Method

Classical Approach

The classical method requires equally likely outcomes. The probability of an event $E$ is:

$P(E) = \frac{\text{number of ways that } E \text{ can occur}}{\text{number of possible outcomes}}$

If $S$ is the sample space, then:

$P(E) = \frac{N(E)}{N(S)}$

where $N(E)$ is the number of outcomes in $E$, and $N(S)$ is the number of outcomes in the sample space.

Example: Classical Probability with M&Ms

• Suppose a sample contains 8 brown, 9 yellow, 6 red, 7 orange, 2 blue, and 2 green candies.

• Probability of yellow: $P(\text{yellow}) = \frac{9}{34}$

• Probability of blue: $P(\text{blue}) = \frac{2}{34}$

Additional info:

• Notes include simulation activities and graphical analysis to reinforce the Law of Large Numbers.

• Tables are used to illustrate probability models and empirical frequency distributions.

• Examples connect theory to practical scenarios, such as dice rolling and daily commutes.