**Probability** is a measure that is associated with how certain we are of the outcomes of a particular experiment or activity. An **experiment** is a planned operation carried out under controlled conditions.

*Today: Overview, Conditional Probability, Independence, Multiplication Rule*

Helpful model:

- Draw a picture (box, circle, etc) – this is everything that ‘could happen’.

Terminology:

- Individual points in the picture are
**outcomes**– the result of one experiment.*Possibilities*. - The whole picture (all outcomes) is called the
**sample space**.*List of all the possibilities.* - An
**event**is a group of one or more outcomes. - The
**probability**of an event is the ratio $\frac{\text{size of event}}{\text{size of sample space}}$. “How much of the total sample space does our event make up?”

### Examples of Experiments

Roll a die.

Choose one person from the class.

A couple is having a baby.

For each example above, what’s the experiment? What are the outcomes? What are some ideas of different events that we might consider? *Hint: You can often come up with an event by making up a question like “what is the probability that blah” — the *blah *is an event.*

Example:

### Example 1

What is the experiment? *Choose a student at random*.

What is the probability of each event? (Find the size of each event, then divide by total possible to find the probability). *The student chosen…*

- Female $P(F)$
- Prefers Drama series $P(D)$
- Doesn’t prefer Reality series $P(R’)$ (note: $R’$ means “$R$ complement”)
- Prefers Reality series $P(R)$
*Question: What’s the connection between the probability of an event and the probability of its complement?*

Combining Events

- Female AND Prefers Drama (
*AND = falls in both events)*$P(F \cap D)$ - Female OR Prefers Drama
*(OR = falls in one event or the other or both)*$P(F \cup D)$

Working with given information (conditional probability)

- What is the probability a student prefers Reality Series?
- What if I tell you the chosen student was Female, now what is the probability that they prefer Reality series?

**Conditional Probability. **$P(A|B)$ means the Probability of A, given B. We think of this as shrinking sample space to only the event B, and asking “how likely is A to occur, *within B*?” $$P(A|B)=\frac{P(A\cap B}{P(B)}$$

### Example 2

### Example 3

What does **independent** mean?

Two events are independent if knowing one of them occurred does not affect the probability that the other occurred.

Defn. Two events $A$ and $B$ are **independent** if $P(A) = P(A|B)$.

Equivalently, if $P(B)=P(B|A)$.

- Back to Example 1: Are the events “Female” and “Prefers Drama” independent?

**Multiplication Rule.** The probability that both $A$ and $B$ occur is equal to the probability that *one* event occurs, times the probability that *the other* event occurs *given that the first event occurred.* $P(A\cap B)=P(B)\cdot P(A|B)$

IF $A$ and $B$ are** independent**, then $P(A\cap B)=P(B)\cdot P(A)$

### Example 4

### Example 5

### Resources on Probability and Statistics

- The Bear in Moonlight – Math With Bad Drawings’ 7-part series on probability (disguised in story form)
- OpenLab course hub for MAT 1372 (Probability and Statistics)
- Introduction to Probability from OpenStax textbook on Probability
- Adjustable spinner (change # of categories and probability of each, then simulate spins)

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