Lec 5 - More probability rules

STAT 1010 - Fall 2022

Learning outcomes

By the end of this lesson you should:

  • Know and understand words like compliment, independent, and the Law of Large Numbers

  • Understand Boole’s inequality

  • Understand the multiplication and addition rules

Law of large numbers (\(\infty\) in practice)

Compliment of \(A\)

\(S - A\)

Addition rule

\(Pr(A \cup B)=Pr(A)+Pr(B)−Pr(A \cap B)\)

Boole’s inequality

\(Pr(A \cup B) \leq Pr(A)+Pr(B)\)

Independent events

Events are said to be independent if the outcome of one event does not impact the probability of another.

Which of these events is independent?

- Removing balls from urns with replacement

- Removing balls from urns without replacement

- Flipping a coin ten times

- Riding in an uber and getting a free meal

- Going into a store and purchasing something from that store

- Suitability of two employees for a job

- Drawing 2 kings from a deck of cards in a row without replacement

3 mins


Conditional probability

When events are not independent conditional probabilities are useful.

We use the notation: \[Pr(Card\: 2\: is\: a\: king∣Card\: 1\: is\: a\: king)\] for this. We use the ∣ as shorthand for “given that” or “conditional on”.

Multiplication rule

\[Pr(A \cap B) = Pr(A)Pr(B∣A)\]

when \(A\) and \(B\) are independent, \(Pr(B|A)\) = ?

  • \(Pr(B|A) = Pr(B)\)

  • \(Pr(A \cap B) = Pr(A)Pr(B)\)

Tree diagrams

Your turn

Click here or the qr code below