ELEC 321

# Tutorial 1

Updated 2017-09-15

## Review

### Total Law of Probability

Given $E_1, E_2,\dotsc,E_n$,

Condition 1: $E_i\cap E_j = \emptyset; i\neq j$

Condition 2 (exhaustive): $\bigcup^n_{i=1}E_i=\Omega$, so $\mathbb P(A)=\sum^n_{i=1}\mathbb P(A\cap E_i)$

### Conditional Probability

Total Law of Property expressed in conditional property:

Example: Binary Communication System

There is a $\frac23$ probability of sending a 0, and $\frac13$ probability of sending a 1. If the bit sent is a 0, there is 0.9 that the output is correct, 0.1 probability of flipped bit. Else if the bit sent is a 1, there is 0.8 chance that that output is correct, but 0.2 chance that the bit is flipped.

Let

So

To find the probability of error, we add the total probability of each error up (total law property)

## Practice

### A1

Given $\mathbb P(A)=\frac13, \mathbb P(B)=\frac13,\mathbb P(A\cap B)=\frac 1{10}$, find:

### A6

Family has two children, and

We want to find $\mathbb P(A\vert B)$

Since they are all equally likely, the probability of each one happening is $\frac14$

And we can know $A\cap B=\{(B,B)\}$ and find the conditional probability.

### A7

There are $n$ people in the room, we want the probability of at least 2 people having same birthday.

Let

The total possibilities of date of birth is $365$ days. and $365-(n-1)$ is number of possibilities with constrain that no people share birthdays.

And

So