ERG 2040 tutorial 1

Zhu Lei

Components of a Probability Model

The probability theory is designed for random experiments. We want to know how likely a certain outcome would appear.

The Sample Space: the totality of the possible outcomes of a random experiment. (S)

An event: a collection of certain sample points, or a subset of the sample space. (E)

Components of a Probability Model

Example 1: Prof. Liew and Prof. Yum do paper-scissors-rock (石头剪子

布 ) to decide who will grade the homework. The rule is: if some one loses, he will do all the works; if it is a draw, they do it together.

First, we pay attention to Prof. Liew’s choice The Sample Space is:

The Event set: any combinations of these three choices Event 1: Prof. Liew chose ‘Scissors’:

E1={Scissors};　　　　　　　　　

Event 2: Prof. Liew did not choose ‘rock’: E2={Paper, Scissors}.

{Paper, Scissors, Rock}

Components of a Probability Model

Example 1:

Prof. Liew and Prof. Yum do paper-scissors-rock (石头剪子布 ) to decide who will grade the homework. The rule is: if some one loses, he will do all the works; if it is a draw, they do it together.

Second, we pay attention to Prof. Liew and Prof. Yum’ choices. The Sample Space is:

It contains 9 possible choices.

The Event set: any combinations of these nine choices Event 1: Prof. Liew chose ‘Scissors’ while Prof. Yum dose not chose ‘Paper’:

E1={ (s,s), (s,r) };

{ (p,p), (p,s), (p,r), (s,p), (s,s), (s,r), (r,p), (r,s), (r,r) }

Components of a Probability Model

Example 1: Prof. Liew and Prof. Yum do paper-scissors-rock (石头剪子

布 ) to decide who will grade the homework. The rule is: if some one loses, he will do all the works; if it is a draw, they do it together.

Next, we pay attention to the outcome of the game. That is who will grade the homework.

The Sample Space is ? A. { Prof. Yum, Prof. Liew}; B. { Prof. Yum, Prof. Liew, Prof Yum & Prof. Liew} C. { Prof. Yum grade homework while Prof. Liew does not, Prof. Liew grade homework while Prof. Yum does not, Both Prof. Yum and Prof Liew grade the homework }

Exclusive Vs. Independent

‘Exclusive’ events are those from the same trial of the same experiment. They cannot happen together.

‘Independent’ events are those from different experiments or different trials of one experiment. The outcome of one event does not influence the others.

Example: E1={Prof. Liew choose paper in the first round}; E2={Prof. Liew choose rock in the first round}; E3={Prof. Yum choose paper in the first round}; E4={Prof. Liew choose rock in the second round};

E1 and E2 are mutually exclusive, because Prof. Liew cannot choose both paper and stone at the same time, although he has two hands.

Mutually Independent Vs. Pairwise Independent

Example: Let Y denote Prof. Yum’s choice. Let L denote Prof. Liew’s choice. Let C denote the outcome of the game; Suppose Prof. Yum and Prof. Liew made their choice randomly.

Is Y independent with C? P(Y∩C)=P(Y)P(C)? Yes

Are they mutually independent? Or P(Y∩L∩C)=P(Y)P(L)P(C)? No

Independent Events

Example: Prof. Liew and Prof. Yum are both very smart guys. After several

months, Prof Liew found out that Prof. Yum likes to use ‘rock’ the most. So he decide to use ‘paper’ to defeat Prof. Yum in the next round. Now is L independent with Y? (Yes)

Prof. Yum found out that Prof. Liew would blink his eyes whenever he want to use scissors. After finding out this, Prof. Yum decide to use rock if Prof. Liew blink his eyes and use paper otherwise. Now is L independent with Y? (No)

• Following Prof. Yum’s strategy, does he still have chance to lose? Yes, he still have chance to lose

Conditional Probability

The conditional probability of A given B is

Example: 100 students took erg2040. 15% of them got Grade A for both

homework and exams; 30% of them got Grade A for homework; 20% of them got Grade A in exams. For a student who did homework very well, what is the probability he got an A in exams?

P(H)=0.3, P(E)=0.2, P(H∩E)=0.15 P(E | H) = P(H∩E)/ P(H)=0.5 This shows doing homework is very helpful!!

Bayes’ rule

The posteriori probability of Bj given A is

Example: It is a peaceful night. You are sleeping.

One of a sudden, the fire alarm rings…………

You remember the fire alarm is 99% accurate,

or P( Alarm| Fire )=99%..............

You got desperate……………..

At this moment, you remember a very important thing that may save your life:

Bayes’ Rule!!

Bayes’ rule

Example: The Fire alarm is 99% accurate, or P( Alarm| Fire )=99%. But it

can be triggered by something else, smoking, candle, etc, So if there is no fire, it can still ring with P( Alarm| no fire)=2%. For a random night, the probability that the dormitory is on fire is very small P( Fire )=0.05%. Given the fire alarm is ringing, what is the probability that there is a fire?

Then you can go back to sleep……

Bayes’ rule

Alarm

Fire Smoking Candle

Bomb

There are many different things that can trigger the alarm. By Bayes’ rule,

we can infer their possibility.

By the Bayes’ rule:

Bernoulli trials

For n trials, the probability of exactly k successes and (n-k) failures:

The outcomes of different trials are independent. win with probability p and lose with probability 1-p for any trial.

The order of outcomes does not matter. ‘win, win, lose’, ‘win, lose, win’ and ‘lose, win, win’ are considered as the

same event with k=2.

Bernoulli trials

Example: We gamble by rolling five dices. If the outcome contains exactly

one ‘6’, you win 1 dollar from me; if not, I win 1 dollar from you. Will you play this with me?

p=1/6, the outcome of one dice is ‘6’. q=1-p=5/6, the outcome of one dice is ‘1’, ‘2’, ‘3’, ‘4’, ‘5’. Pr{ you win } = p{exactly one dice is ‘6’}

Bernoulli trials

Example: We gamble by rolling five dices. If the outcome contains exactly

two ‘1’s and two ‘2’s, you win one dollar from me. If it contains exactly three ‘2’s, I win one dollar from you.

p1=1/6, the outcome of one dice is ‘1’.

p2=1/6, the outcome of one dice is ‘2’.

A question from homework

Three couples (husbands and their wives) must sit at a round table in such a way that no husband is placed next to his wife. How many configurations exist?