Lecture 4: Probability

From lectures: 4.1, 4.2, 4.3

What is probability?

A numerical measure of how likely an event is to occur.

What does a probability of 0 mean?

The event is impossible.

What does a probability of 1 mean?

The event is certain.

What is a statistical experiment?

A process that generates outcomes determined by chance.

What is a random experiment?

An experiment where repeating it can produce different outcomes.

What is a sample space (S)?

The set of all possible outcomes in an experiment.

What is a sample point?

A single outcome in the sample space.

What is an event?

A collection of sample points.

Rule: Probability range

0 ≤ P(E) ≤ 1.

Rule: Sum of all probabilities?

Must equal 1.

Classical method of assigning probability?

All outcomes are equally likely.

Relative frequency method?

Uses data: probability = frequency ÷ total.

Subjective method?

Based on judgment or belief.

What is the complement of A?

All outcomes NOT in A (Aᶜ).

Formula for complement?

P(Aᶜ) = 1 – P(A).

What is A ∪ B (union)?

Outcomes in A or B or both.

Formula for P(A ∪ B)?

P(A) + P(B) – P(A ∩ B).

What is A ∩ B (intersection)?

Outcomes in both A and B.

What are mutually exclusive events?

Events that cannot happen at the same time.

Rule for mutually exclusive events?

P(A ∩ B) = 0.

What are independent events?

One event does NOT affect the other.

Conditional probability formula?

P(A | B) = P(A ∩ B) / P(B).

What does Bayes’ Theorem find?

Updated (“next”) probabilities after new information.

What are independent events?

Events that do NOT affect each other’s probability. One happening doesn’t change the chance of the other happening. Example: Being left-handed is independent of working in a bank.

Formula for independent events?

P(A AND B) = P(A) × P(B)

What are dependent events?

Events where the probability of one changes depending on whether another event happened. Uses conditional probability. Example: Employment may depend on having higher education.

Formula for dependent events?

P(A | B) = P(A AND B) / P(B) or P(B | A) = P(A AND B) / P(A)

How to read P(A | B)?

“Probability of A given B.”

Income & University Survey example: Are “Income ≥ 40k” and “Attended University B” independent?

No, because P(Income ≥ 40k) = 0.25 and P(Income ≥ 40k | Uni B) = 0.333 → not equal → dependent

Superpower Survey example: P(male | fly)?

26/38 = 0.68

Superpower Survey example: P(fly | male)?

26/48 = 0.541

Are coin tosses and dice rolls independent?

Yes. Example: Tossing 2 coins → P(T and T) = 1/2 × 1/2 = 1/4; Rolling 2 dice → P(3 and 3) = 1/6 × 1/6 = 1/36

Socks experiment example (5 colors, with replacement): P(red and red)?

1/5 × 1/5 = 1/25

Coin + die experiment example: P(heads AND 3)?

1/2 × 1/6 = 1/12

How to check if events are independent?

Check if P(A AND B) = P(A) × P(B) OR P(A | B) = P(A); if equal → independent, if not → dependent

Replacement vs No Replacement:

With replacement → usually independent; without replacement → usually dependent

Mutually Exclusive vs Independent:

Mutually exclusive: cannot happen together (P(A AND B)=0); Independent: can happen together but one does NOT affect the other

Key words hinting dependence:

“Given that”, “after”, “if”, “depends on” → usually dependent; “does not affect”, “simultaneously” → usually independent