# Lecture 13: conditional probability

• Reading: Cameron 1.10, chapter 2

• Last semester's notes

• Definitions: independent events, conditional probability
• Modeling with conditional probability
• Statement of Bayes's rule

## Independence

Definition: Two events $$A$$ and $$B$$ are independent if $$Pr(A \cap B) = Pr(A)Pr(B)$$.

Important: Do not assume events are independent unless given a good reason to do so.

Example: Suppose I roll two 6-sided dice. For either die, the probability of getting any number from 1...6 is 1/6. What is the probability of getting a pair of twos?

Answer: it could be anything from 0 to 1/6. The dice could be taped together in a way that it is impossible to get (2,2) while still making the probability of any given roll for either die to be 1/6.

If we are also told that the two rolls are independent, then we can conclude that \begin{aligned} Pr(\{(2,2)\}) &= Pr(\{(2,n)\mid n \in \{1,\dots,6\}\} \cap \{(n,2) \mid n \in \{1,\dots,6\}\}) \\ &= Pr(\{(2,n)\mid n \in \{1,\dots,6\}\})\cdot Pr(\{(n,2) \mid n \in \{1,\dots,6\}\}) \\ &= (1/6)(1/6) \\ \end{aligned}

## Conditional probability

Definition: If $$A$$ and $$B$$ are events, then the probability of A given B, written $$Pr(A|B)$$ is given by $Pr(A|B) ::= \frac{Pr(A \cap B)}{Pr(B)}$ Note that $$Pr(A|B)$$ is only defined if $$Pr(B) \neq 0$$.

Intuitively, $$Pr(A|B)$$ is the probability of $$A$$ in a new sample space created by restricting our attention to the subset of the sample space where $$B$$ occurs. We divide by $$Pr(B)$$ so that $$Pr(B|B) = 1$$.

Note: $$A|B$$ is not defined, only $$Pr(A|B)$$; this is an abuse of notation, but is standard.

## Alternative definition of independence

Conditional probability can be used to give an equivalent definition of independence:

Claim: $$A$$ and $$B$$ are independent if and only if $$Pr(A|B) = Pr(A)$$.

This is perhaps a more intuitive notion of independence: if I tell you that $$B$$ happened, it doesn't change your estimate of the probability that $$A$$ happens.

Proof: (⇒) Suppose $$A$$ and $$B$$ are independent. We wish to show $$Pr(A|B) = Pr(A)$$. Well, by definition, $$Pr(A|B) = Pr(A \cap B)/Pr(B)$$. Since $$A$$ and $$B$$ are independent, $$Pr(A \cap B) = Pr(A)Pr(B)$$. Plugging this in, we see that $$Pr(A|B) = Pr(A)Pr(B)/Pr(B) = Pr(A)$$.

(⇐) Suppose $$Pr(A|B) = Pr(A)$$. Then $$Pr(A) = Pr(A \cap B)/Pr(B)$$. Multiplying both sides by $$Pr(B)$$ gives the desired result.

## Bayes's Rule

Bayes's rule is a simple way to compute $$P(A|B)$$ from $$P(B|A)$$.

Claim: (Bayes's rule): $$P(A|B) = P(B|A)P(A)/P(B)$$.

Proof: left as exercise.

Example: See next lecture

## Probability trees

Using conditional probability, we can draw a tree to help discover the probabilities of various events. Each branch of the tree partitions part of the sample space into smaller parts.

For example: suppose that it rains with probability 30%. Suppose that when it rains, I bring my umbrella 3/4 of the time, while if it is not raining, I bring my umbrella with probability 1/10. Given that I bring my umbrella, what is the probability that it is raining?

One way to model this problem is with the sample space

$S = \{raining (r), not raining (nr)\} \times \{umbrella (u), no umbrella (nu)\} = \{(r,u), (nr,u), (r,nu), (nr,nu)\}$

Let $$R$$ be the event "it is raining". Then $$R = \{(r,u), (r,nu)\}$$. Let $$U$$ be the event "I bring my umbrella". Then $$U = \{(r,u), (nr,u)\}$$.

The problem tells us that $$Pr(R) = 3/10$$. It also states that $$Pr(U|R) = 3/4$$ while $$Pr(U|\bar{R}) = 1/10$$. We can use the following fact:

Fact: $$Pr(\bar{A}|B) = 1 - Pr(A|B)$$. Proof left as exercise.

to conclude that $$Pr(\bar{U}|R) = 1/4$$ and $$Pr(\bar{U}|\bar{R}) = 9/10$$.

We can draw a tree:

We can compute the probabilities of the events at the leaves by multiplying along the paths. For example, $Pr(\{(r,u)\}) = Pr(U \cap R) = Pr(R)Pr(U | R) = (3/10)(3/4) = (9/40)$

To answer our question, we are interested in $$Pr(R|U) = Pr(U \cap R)/Pr(U) = (9/40)/(3/10) = 3/4$$.

Note we could also answer this using Bayes's rule and the law of total probability; it would amount to exactly the same calculation. The tree just helps organize all of the variables.