# Lecture 5: conditional probability

## 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.

## 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 $$Pr(R|U) = Pr(U \cap R)/Pr(U)$$. We know $$U = \{(u,r), (u,nr)\}$$. We can compute $$Pr(U)$$ using the third axiom; $$Pr(U) = Pr(\{(u,r)\}) + Pr(\{(u,nr)\}) = (3/10)(3/4) + (7/10)(1/10)$$. We can then plug this in to the above formula to find $$Pr(R|U)$$.

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

## 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: Write down the definitions; proof left as exercise.

## Law of total probability

Claim: (law of total probability) If $$A_1, \dots, A_n$$ partition the sample space $$S$$ (that is, if $$A_i \cap A_j = \emptyset$$ for $$i \neq j$$ and $$S = \cup A_i$$), then

$Pr(B) = \sum_{i} Pr(B|A_i)Pr(A_i)$

Proof sketch: Write $$B = \cup_{i} (B \cap A_i)$$. Apply third axiom to conclude $$Pr(B) = \sum_{i} Pr(B \cap A_i)$$. Apply definition of $$Pr(B | A_i)$$.