business statistics l2

7/27/2019 Business Statistics L2

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Business Statistics

Fall, 2013Probability


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Lecture outline

Random experiment

What is probability

Sample space and events Probability Postulates

Probability Rules


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Suppose that a process that could lead to two or moredifferent outcomes is to be observed, so there isuncertainty beforehand as to which outcome will occur.

- A coin is thrown

- A die is rolled- A consumer is asked which of two products she

prefers

- The daily change in an index of stock market prices

is observed

Each of these examples involves a random experiment.


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Random Experiment

A random experiment is a process leading to atleast two possible outcomes with uncertainty as

to which will occur.


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Language of probability helps us to makeprecise statements about the nature of ouruncertainty.

Indeed, we can think of probability as thelanguage in which we discuss uncertainty.

We will now turn to develop a formalstructure for making probability statements.


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What is probability?

Subjective probability

Subjective probability is personal. Personal

hunch, based on the knowledge or experiencethe person has and the way the person

interprets it, his intuition.


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Subjective probability

- Common form of probability in business and policymaking

What is the probability our product will be successful in the market

What is the probability that the S&P 500 price index will go uptomorrow

Corporate executives face decisions as to whether to makepotentially lucrative investments in countries that have unstablepolitical climates. Either formally or informally, it is necessary toassess the likelihood of the nationalization of the corporations

assets. Such assessment must surely be subjective.

- Should be used with a high level of skepticism


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Relative frequency interpretation of probability

Let be the number of occurrences of eventAinNrepeated trials (outcomes are not interdependent),

Proportion of occurrences of eventAin Ntrials =

Now suppose N becomes large

Probability that A occurs =

AN

N

AN

lim AN

N

N


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Relative frequency

Probability is viewed as a result of long-run

convergence. In the short runas in a

casino, we can see streaks of winning, but

in the long run, with enough hands of

blackjack, you should expect losses.


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Classical interpretation of probability

- Very similar o the relative frequency view

- Assumes all possible outcomes are equally likely to occur

- Important idea: we can obtain a probability from a logical

reasoning about the process, without performing the

random experiments for a large number of times


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Classical

Probability that an event will occurnumber of outcomes in the sample space that satisfy the event

number of all possible outcomes in the sample space

Requirements: - can count the total number of possibleoutcomes in the sample space

- can count the number of ways an eventcan occur


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An example particularly relevant in Xiamen

Mid-Autumn FestivalXiamen Bo Bing

Rolling six dice together and win soaps, tissues, and

toothpastes

Rolling six dice together, how many possible outcomes?

S= {(1,2,3,4,5,6), (2,1,3,4,5,6), (3,2,1,4,5,6), (4,4,4,4,1,1), }

Actually 46656 possible outcomes. We will learn how tocount the total number of possible outcomes later.


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Define event A = to get Winner Wearing Blossoms (four fours and twoones) when rolling six dice together.

A={(4,4,4,4,1,1), (4,1,1,4,4,4), (4,4,1,1,4,4), }

How many possible outcomes in this set? Actually 15.We will learn how to count the outcomes later.

So you see how hard it is to get Winner Wearing Blossoms?

P(A) = 15/46656 = 0.000322 = 0.03%

If there were items to give away but is marked for Winner WearingBlossoms,sometimes people have to play long time to get the diceturn out the streak we want.


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Mutually Exclusive Events

Mutually exclusive events cannot logically happening at thesame time.

e.g., (i) A flipped coin coming up heads and the same coincoming up tails at the same time

(ii) The two events it rained the whole day on Mondayand it did not rain at all on Mondayare mutually exclusive

events

Name a pair of mutually exclusive events to your neighbor!


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Lets look at the events A and A complement

A A S

A A

Notice that A an A complement are MECE (me-see),

or mutually exclusive and collectively exhaustive.


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Lets do some quick practice to get familiar with the ideas of

complement, intersection, and unions.


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Some immediate consequences of the three

postulates

(i) If the sample space S consists of n equally

likely basic outcomes, then each of these has

probability 1/n; that is

1( )

iP O

n

This follows from the third postulate. If P(Oi) is the same for

each basic outcome and the summation of the probabilities

of all basic outcomes is equal to 1, then P(Oi) must be 1/n.

For example, if a fair die is rolled, the probability for each of

the six basic outcomes is 1/6.


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(iii) Let A and B be mutually exclusive events.

Then the probability of their union is the sum of

their individual probabilities; that is

More generally, if E1

, E2

, , Ek

are mutually

exclusive events

( ) ( ) ( )P A B P A P B

1 1 1 2( ) ( ) ( ) ( )K KP E E E P E P E P E


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This result is a consequence of the second postulate. The

probability of the union of A and B is

Where the summation extends over all the basic outcomes

in the union of A and B. But since A and B are mutually

exclusive, no basic outcome can belong to both, so the

above formula can be broken down into the sum of twoparts:

( ) ( )i

A B

P A B P O

( ) ( ) ( ) ( ) ( )i i i

A B A B

P O P O P O P A P B


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Class practice

A hamburger chain found that 75% of all

customers use mustard, 80% use ketchup, and

65% use both. What is the probability that a

particular customer will use at least one of

these?


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