Probability Basics for Statistics

Manual: General · Subject: Statistics

Develop the probability ideas needed to reason about uncertainty and inferential methods.

Probability as a Model of Uncertainty

From experiments to probability

Probability quantifies the chance of outcomes in a random process. In statistics, probability is used as a model for how samples vary and how likely observed results are under a hypothesized process.

Fundamental ideas

Sample space

The set of all possible outcomes.

Event

A subset of the sample space.

Complement

The event that something does not happen.

Independent events

Events where knowing one does not change the probability of the other.

Useful probability rules

Rule	Formula	Meaning
Complement rule	$P(A^c)=1-P(A)$	Probability an event does not occur
Addition rule	$P(A\cup B)=P(A)+P(B)-P(A\cap B)$	Probability of $A$ or $B$
Multiplication rule	$P(A\cap B)=P(A)P(B\mid A)$	Probability of both events

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Conditional probability

Conditional probability updates probability after new information: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ when $P(B)>0$ .

If $P(A)=0.3$ , what is $P(A^c)$ ?

0.7 0.3 1.3 0

What does it mean for two events to be independent?

Solving a basic probability problem

1

Step 1: Define the sample space and event(s).
2

Step 2: Choose the rule that matches the structure of the question.
3

Step 3: Compute the probability carefully using complements, unions, or conditionals.
4

Step 4: Check whether the answer is reasonable and between 0 and 1.

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Probability Basics for Statistics

Probability as a Model of Uncertainty

From experiments to probability

Solving a basic probability problem