Probability

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?”

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

1. Female $P(F)$
2. Prefers Drama series $P(D)$
3. Doesn’t prefer Reality series $P(R’)$ (note: $R’$ means “$R$ complement”)
4. Prefers Reality series $P(R)$
   • Question: What’s the connection between the probability of an event and the probability of its complement?
5. Female AND Prefers Drama (AND = falls in both events) $P(F \cap D)$
6. Female OR Prefers Drama (OR = falls in one event or the other or both) $P(F \cup D)$

Conditional probability

Definition: 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 cont’d – Working with given information (conditional probability). Use the data from Example 1 to find each probability.

1. What is probability that a student prefers Reality Series, given that they are Female $P(R|F)$? That is, if I tell you the chosen student was Female, now what is the probability that they prefer Reality series? Is this the same as the probability $P(R)$ that a student prefers Reality Series (with no information given about their gender)?
2. What is the probability that student is Male, given that they like Drama Series?
3. What is the probability that a randomly selected Female student likes Drama Series?
4. What is the probability that a student is both Male and likes Comedy series, if they are selected randomly from all students who are either Male or who like Comedy Series?

Independent Events

What does independent mean in discussions about probability?

Two events are independent if knowing one of them occurred does not affect the probability that the other occurred. Technically, this means:

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

Example.

Multiplication Rule

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 $P(A\cap B)=P(B)\cdot P(A)$