# 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

# Compliment of $A$

$S - A$

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

# 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

03:00

# 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, $Pr(B|A)$ = ?

• $Pr(B|A) = Pr(B)$

• $Pr(A \cap B) = Pr(A)Pr(B)$