## MA2102

# Probability and Statistics

#### Lecture-7

### Conditional Probability:

Probabilities of events get modified when we have some partial information about the outcome.

Consider the random experiment of rolling a fair die, sample space $\Omega=\{1,2,3,4,5,6\}$ and let us be interested in the following events here.

Let $A=\{2,3,5\}\leftarrow$ die landing on a prime number

$B=\{2,4,6\}\leftarrow$ die landing on a even number

$C=\{1,3,5\}\leftarrow$ die landing on odd number

Intuitively we can see that $P(A$ given that $B$ already happened$)=\frac{1}{3}$

Because when it is given that $B$ already happened, we think of only 2, 4, 6 as possible outcomes, so $B$ kind of becomes new reduced sample space, and for $A$ only 2 is favourable for being prime.

And $P(A$ given that $B$ already happened$)=\frac{1}{3}\ne \frac{1}{2}=P(A)$

That is when we know that B happened probability of A got reduced from $\frac{1}{2}$ to $\frac{1}{3}$

similarly we can argue that $P(A$ given that $C$ already happened$)=\frac{2}{3}\ne \frac{1}{2}=P(A)$

That is when we know that C happened probability of A got increased from $\frac{1}{2}$ to $\frac{2}{3}$

So when partial information about outcome is available , we might have to update the probability of event.

**Question:** How to update the probability of an event using partial information?

To get an intuition, let us focus on random experiment with finite sample space and equally likely outcomes.

Let $\Omega$ be a sample space, and $A,B$ ar events. and we are interested in determining probability of $A$, when we have information that $B$ happened.

So, $B$ is now, new reduced sample space and only outcomes of $A$ that concern us are those (if any) that are also outcomes from $B$

(i.e outcomes in $A\cap B$)

So it makes sense to define $P(A$ given that $B$ already happened)$=\frac{|A\cap B|}{|B|}$

and $=\frac{|A\cap B|}{|B|}=\frac{\frac{|A\cap B|}{|\Omega|}}{\frac{|B|}{|\Omega|}}=\frac{P(A\cap B)}{P(B)}$

So, intuition suggests that when $B$ already happened,new probability of $A$ should be updated as $\frac{P(A\cap B)}{P(B)}$ (in terms of old probabilities)

Let us check this rule of updating probability with above die example that we considered.

$P(A$ given that $B$ happened$)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{1}{6}}{\frac{1}{2}}=\frac{1}{3}$ and $P(A$ given that $C$ happened$)=\frac{P(A\cap C)}{P(C)}=\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3}$

Now let us show that the updated probabilities are indeed representing valid probabilities.

Let $(\Omega,\mathscr{F},P)$ be probability space with $P(B)>0$

Define a set function, $P_B:\mathscr{F}\rightarrow \mathbb{R}$, $P_B(A)=\frac{P(A\cap B)}{P(B)}$

*claim*: $P_B$ is a probability (measure) function.

Here we need to check three axioms of probability for $P_B$ function.

- $P_B(A)=\frac{P(A\cap B)}{P(B)}\ge0~\forall A \in \mathscr{F}$ ($\because P$ already probability function $\Rightarrow P(A\cap B)\ge0 ,~P(B)>0$ by Axiom 1)

So Axiom 1 for $P_B\checkmark$

- $P_b(\Omega)=\frac{P(\Omega \cap B)}{P(B)}=\frac{P(B)}{P(B)}=1$ ($\because B\subseteq \Omega \Rightarrow \Omega\cap B= B$ )

So Axiom 2 for $P_B\checkmark$

- Let $A_1,A_2,A_3,...,A_n,...$ are mutually exclusive events in $\mathscr{F}$

$P_B(\bigcup_{i=1}^{\infty}A_i)=\frac{P((\bigcup_{i=1}^{\infty}A_i)\cap B)}{P(B)}$

$=\frac{P(\bigcup_{i=1}^{\infty}(A_i\cap B))}{P(B)}$ ($\because$ generalized distributive laws)

$=\frac{\sum_{i=1}^{\infty}P(A_i\cap B)}{P(B)}$ ($\because A_1\cap B, A_2\cap B, A_3\cap B,.....$also mutually exclusive and Axiom 3 for $P$ )

$=\sum_{i=1}^{\infty}\frac{P(A_i\cap B)}{P(B)}=\sum_{i=1}^{\infty}P_B(A_i)$ ($\because$ definition of $P_B$)

So Axiom 3 for $P_B\checkmark$

$\therefore P_B$ is a probability measure

For any $A\in \mathscr{F}, P_B(A)$ is calledof $A$ given that $B$ already happened.*conditional probability*

In practice $P_B(A)$is denoted by $P(A/B)$ and read it as 'probability of A given B'

$P(A/B):=\frac{P(A\cap B)}{P(B)}~,P(B)>0$

__Example 1:__A family has two children. What is the conditional probability that both are boys given that(i) family has a boy? (ii) elder one is boy?

*solution*: Let sample space $\Omega=\{(b,b),(b,g),(g,b),(g,g)\}$

Let $A$ denote the event that family having both boys, i.e $A=\{(b,b)\}\Rightarrow P(A)=\frac{1}{4}$,

$B$ denote the event that family having a boys, i.e $B=\{(b,b),(b,g),(g,b)\}\Rightarrow P(B)=\frac{3}{4}$,

$C$ denote the event that family having elder one boy, i.e $C=\{(b,b),(b,g)\}\Rightarrow P(C)=\frac{1}{2}$

(i). $P(A/B)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$

(ii). $P(A/C)=\frac{P(A\cap C)}{P(C)}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$

*Note*: Most people don't see that (i) and (ii) are actually different, i.e people approach (i) as (ii) and come up with answer $\frac{1}{2}$,

please understand the difference.