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’.


  • 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 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