Lecture 9 & 10 PDF

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Lecture 9 & 10...

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Lecture 9 & 10 – Numerical Descriptive Measures NOTES Probability Assigning probabilities to events 

Random experiment is the process or course of action whose outcome is uncertain.

Sample space 

A list of all possible outcomes of a random experiment: S = {…..}

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An event is any collection of one or more possible outcomes.

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Simple events are individual outcomes.

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The objective is to determine P(A), the probability that event A will occur.

Approaches to assigning probabilities 1. Classical approach – based on counting “possible” and “favourable” events. 2. Relative frequency – assigning probabilities based on experimentation or historical data. 3. Subjective approach – assigning probabilities based on the assignor’s (subjective) judgement. Classical approach 

If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome. It is necessary to determine the number of possible outcomes.

Relative frequency approach 

E.g. as 3 desktops are sold on 12 days out of the 30 days, therefore, there is 40% chance that 3 desktops will be sold on any given day.

Subjective approach 

In the subjective approach, the probability is defined as the degree of belief that we hold in the occurrence of an event.

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E.g. probability of precipitation is defined in different ways by different forecasters, but basically, it’s a subjective probability based on past observations combined with current weather conditions, computed through a model.

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POP 60% - based on current conditions, there is a 60% chance of rain.

Interpreting probability 

All will be interpreted in the relative frequency approach.

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E.g. a government lottery game where 6 numbers (of 49) are picked. The classical approach would predict the probability for any one number picked as 1/49 = 2.04%. This is interpreted that in the long run each number will be picked 2.04% of the time.

Assigning probabilities

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Probability of an event – the probability P(A) of event A is the sum of the probabilities assigned to the simple events contained in A.

Complement, union and intersection of events 

We study methods to determine probabilities of events that result from combining other events in various ways.

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There are several types of combinations and relationships between events -

Complement of an event

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Intersection of events

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Union of events

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Mutually exclusive events

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Dependent and independent events

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Conditional event

Probability of combinations of events

Complement of an event

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the complement of event A is defined to be the event consisting of all sample points that are ‘not in A’.

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complement of A is denoted by

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P(A) + P(Ac) = 1

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E.g. possible tosses of 2 coins S = {(H,H), (H,T), (T,H), (T,T)} -

Let A = observing at least one head = {(H,H), (H,T), (T,H)}

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Ac = {(T,T)}

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P(A) = ¾, P(AC) = ¼

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P(A) + P(AC) = P(S) = 1

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P(AC) = 1 – P(A)

Intersection of two events 

The intersection of events A and B is the set of all sample points that are in both A  B.

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The intersection is denoted by A  B

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The joint probability of A and B is the probability of the intersection of A and B. i.t. P(A  B)

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E.g. possible tosses of 2 dices S = {(1,1), (1,2),…, (6,6)} -

Let A = tosses where first toss is 1 = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)}

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B = tosses where the second toss is 5 = {(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)}

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The intersection, A  B = {(1,5)}. The joint probability of A and B is the probability of the intersection of A and B, i.e. P(A  B) = 1/36

Union of two events

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The union of two events A and B, is the event containing all outcomes that are in A or B or both.

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Union of A and B is denoted by A  B

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E.g. possible tosses of 2 dices S = {(1,1), (1,2),…, (6,6)} -

Let A = tosses where first toss is 1 = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)}

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B = tosses where the second toss is 5 = {(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)}

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Union of A and B, A  B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,5), (3,5), (4,5), (5,5), (6,5)}

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The probability of A  B = 11/36, P(A  B) = P(A) + P(B) – P(A  B)

Mutually exclusive events 

When two events are mutually exclusive (the events have no outcomes in common), their joint probability is 0.

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E.g. possible tosses of 2 dices S = {(1,1), (1,2),…, (6,6)} -

A = tosses totalling 9 = {(3,6), (6,3), (4,5), (5,4)}

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B = tosses totalling 11 = {(5,6), (6,5)}

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Therefore, P(A  B) = 0

Example The number of spots turning up when a 6-sided die is tossed. a) Define the sample space for this random experiment and assign probabilities to the simple events. S = {1, 2, 3, 4, 5, 6} As each simple event is likely to occur, thus, P(1) = P(2) =…= P(6) = 1/6

b) Find P(A) = P(the number observed is at most 2) P{1,2} = P(1) + P(2) = 1/6 + 1/6 = 2/6 = 0.33 c) Find P(AC) AC = {3,4,5,6} P(AC) = P(3) + P(4) + P(5) + P(6) = 4/6 = 0.67

d) Are events A and C mutually exclusive? Events A and C are mutually exclusive because they cannot occur simultaneously.

e) Find P(A  C)

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A = {1,2}, C = {4}, A  C = {1, 2, 4} P(A  C) = P(1, 2, 4) = 1/6 + 1/6 + 1/6 = 3/6 = 0.5 f) P(A  B) A = {1,2}, B = {2, 4, 6}, A  B = {2} P(A  B) = P(2) = 1/6 = 0.167 g) Find P(A  B) A = {1,2}, B = {2, 4, 6}, A  B = {1, 2, 4, 6} P(A  B) = P(1, 2, 4, 6) = 1/6 + 1/6 + 1/6 + 1/6 = 4/6 = 0.67 h) Find P(CB) C = {4}, B = {2, 4, 6} P(CB) = P(the number is 4 given that an even number has occurred) = 1/6 + 1/6 = 1/3 = 0.33 Joint, marginal and conditional probability 

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Intersection -

The intersection of event A and B is the event that occurs when both A and B occur.

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The intersection of events A and B is denoted by (A  B).

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The joint probability of A and B is the probability of the intersection of A and B, and is denoted by P(A  B).

Marginal probabilities -

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Marginal probabilities are computed by adding across rows and down columns which is calculated in the margins of the table.

Conditional probability -

It is used to determine how two events are related which is the probability of one even given the occurrence of another related event.

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Conditional probabilities are written as P(AB) and read as the probability of A given B: P(AB) = P(A  B)/P(B) or P(BA) = P(B  A)/P(A)

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As the occurrence of another event changed the probability of the other event, the two events are related and are called ‘dependent events’.

Independence

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the probability of one event is not affected by the occurrence of the other event.

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E.g. P(AB) = P(A)

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E.g. A2 and B2 are dependent. That is, the probability of event B2 is affected by the occurrence of the event A2. This means that Fund outperforms the market does depend on whether the manager graduated from a top-20-MBA program.

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Independent events and mutually exclusive event are not the same.

Rules of probability 

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Complement rule -

Each simple event must belong to either A or its complement, A C

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The sum of the probabilities assigned to all simple event is one

Addition rule -

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For any two events A and B  P(A  B) = P(A) + P(B) – P(A  B)

Multiplication rule -

Intersection and when A and B  P(A  B) = P(BA) P(A), = P(AB) P(B)

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are independent  P(A  B) = P(A)P(B)...