Section 5.1

Probability Rules

Understand the fundamental rules of probability, sample spaces, and how to build valid probability models.

1

Probability as a Measure of Likelihood

Definition

Probability is the long-term relative frequency of an outcome. It tells us how likely an event is to occur when we repeat an experiment many times.

LogicLens: The Law of Large Numbers

As the number of repetitions of a probability experiment increases, the proportion of times a specific outcome occurs approaches the true probability.

Example: Flipping a Fair Coin
10 flips → might get 7 heads (70%) — doesn't mean P(heads) = 0.7
100 flips → might get 53 heads (53%) — getting closer
10,000 flips → will be very close to 50% — approaches true probability

Law of Large Numbers Simulator

Flip a fair coin repeatedly. Observe how the proportion of heads (the empirical probability) converges to the true probability (0.50) as the number of trials increases.

Total Flips
0
Heads Count
0

Proportion of Heads vs. Number of Flips

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2

The Rules of Probability

1

Range Rule

Probabilities are never negative and never exceed 1 (100%).

2

Sum Rule

The sum of probabilities of all possible outcomes must equal exactly 1.

Impossible Event

An event that cannot occur (e.g., rolling a 7 on a standard die)

Certain Event

An event that will definitely occur (e.g., rolling 1-6 on a die)

3

Sample Spaces and Events

Sample Space (S)

The set of all possible outcomes of an experiment.

Example: Rolling a die

Event (E)

A subset of the sample space — one or more outcomes we're interested in.

Example: Rolling an even number

LogicLens: Classical Probability Method

When all outcomes are equally likely, use this formula:

= number of ways E can occur
= total outcomes in sample space
Example: P(even on a die) =

Sample Space Calculator

For repeated independent trials, the sample space size is:

Sample Space Size
Examples:
  • • 3 coin flips: outcomes
  • • 2 dice rolls: outcomes
  • • Family with 4 children: gender combinations

Coin Flip Experiment

Outcomes: 2 (Heads/Tails)

2
Total Outcomes:
H
H
H
T
T
H
T
T
4

Probability Models

Definition

A probability model is a complete list of all possible outcomes along with their corresponding probabilities.

Outcome123456Sum
P1/61/61/61/61/61/61 ✓

LogicLens: Validating a Model

To verify a probability model is valid, check:

All

Unusual Events

An event is considered "unusual" if its probability is less than (5%).

Example: If P(E) = 0.03, the event is unusual because 3% < 5%

LogicLens Practice

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