Welcome to MM305

Unit 3 Seminar

Prof Greg

Probability Concepts and

Applications

The Basics of Probability• Events• Outcomes• Probability Experiment• Sample Space

Probability Basics• Experiment: Rolling a single die• Sample Space: All possible outcomes from

experiment• S = {1, 2, 3, 4, 5, 6}

• Event: a collection of one or more outcomes (denoted by capital letter)• Event A = {3}• Event B = {even number}

• Probability = (number of favorable outcomes) / (total number of outcomes)• P(A) = 1/6• P(B) = 3/6 = ½

More Probability Basics • Probability will always be between 0 and 1. It

will never be negative or greater than 1.

• Complement of an event: All outcomes that are not included in the Event of interest. • If A = {3} then the “not A” or A’ = {1, 2, 4, 5, 6}.

A’ is everything but 3

• The sum of the simple probabilities for all possible outcomes of an activity must equal 1

The Basics of ProbabilityThree ways to calculate probability:Classical Probability: Proportion of times that an event can be theoretically expected to occur. For outcomes that are equally likely to occur, Probability of Event X= (total number of favorable outcomes for event X)

(total number of possible outcomes)This is the standard way to calculate probability

Relative Frequency Probability: Proportion of times that a probability is expected to occur over a large number of trials. For a very large number of trials, Probability of Event X= (total number of trials for event X)

(total number of trials)

Subjective Probability: Probabilities estimated by making an educated guess; based solely on belief that the event will happen

More Basics Concepts of ProbabilityIndependent EventsTwo events are said to be independent if the outcome of the second event is not affected by the outcome of the first event. They cannot influence or affect each other.

Mutually Exclusive Events Two events are said to be mutually exclusive if they cannot occur at the same time.

Compound Probability AND P(A and B) = P(A)*P(B) when the events are independent P(A and B) = P(A) + P(B) – P(A or B) when the events are dependent

Compound Probability OR P(A or B) = P(A) + P(B) when the events are mutually exclusive P(A or B) = P(A) + P(B) – P(A and B) when the events are not mutually exclusive

Conditional Probability P(B | A), event B given that event A has occurred ( P(B | A) ≠ P(A | B) ) P(B | A) = P(B) and P(A|B) = P(A) when events are independent

Mutually Exclusive EventsEvents are said to be mutually exclusive if only one of the events can occur on any one trial

Tossing a coin will result in either a head or a tail

Rolling a die will result in only one of six possible outcomes

Probability: Tying it all together

0.00%(A)

0.01-0.09%(B)

≥0.10%(C)

Total

0-19 (D)

142 7 6 155

20-39 (E)

47 8 41 96

40-49(F)

29 8 77 114

Over 60(G)

47 7 35 89

Total 265 30 159 454

Blood Alcohol Level of Victim

Venn Diagrams

P (A) P (B)

Events that are mutually exclusive

P (A or B) = P (A) + P (B)

Events that are not mutually exclusive

P (A or B) = P (A) + P (B) – P (A and B)

P (A) P (B)

P (A and B)

Random Variables

Discrete random variablesDiscrete random variables can assume only a finite or limited set of values Continuous random variablesContinuous random variables can assume any one of an infinite set of valuesAlways define what your random variable represents!

Let X = number of people, companies, computers, hours, etc.

A random variable assigns a real number to every possible outcome or event in an experiment

Numerical Descriptors of a Discrete Probability Distribution

General Formulas for mean and variance:

Mean (Expected Value) µ = Σ (x*P(x) )

Variance σ2 = Σ ( (x- µ)2 * P(x) )

Standard Deviation = σ = √σ2

for all possible values of x

QM for Windows : Select Statistics

QM for Windows : Select Data Analysis

QM for Windows : Select # Values, Data Type

QM for Windows : Enter Values; Press Solve

QM for Windows : Table with Mean, Variance

QM for Windows : Select Window then Graph

Excel QM : Select Probability Distribution

Excel QM : Select # Values, Data Type

Excel QM : Enter Values => Mean, Variance

Binomial Distribution1: The number of trials n is fixed. 2: Each trial is independent. 3: Each trial represents one of two outcomes ("success" or "failure"). 4: The probability of "success" p is the same for each outcome.

If these conditions are met, then X has a binomial distribution with parameters n and p, denoted X~B(n, p).

The Binomial DistributionEach trial has only two possible outcomesThe probability stays the same from one trial to the nextThe trials are statistically independentThe number of trials is a positive integer

Expected Value (Mean) and Variance of The Binomial Distribution

Mean (Expected Value) µ = E(x) =n*p

Variance σ2 = n* p *(1- p)

Standard Deviation = √σ2 = √n* p *(1- p)

Where n = number of trials x = number of successes p = probability of success (1- p) = probability of failure

Binomial DistributionSuppose 12% of telemarketers make a sale on a cold call, what is the probability if 10 telemarketers make a cold call that 3 of them will make a sale?

Identify what we know:n= 10 x=3p=0.12 q=1-0.12=0.88

Excel Function: BINOMDIST

P(X=3) = BINOMDIST(3,10,0.12,FALSE) = 0.0847P(X<=3) = BINOMDIST(3,10,0.12,TRUE) = 0.9761P(X>3) = 1 - P(X<=3) = 1 - 0.9761 = 0.0239

E(X)= n*p= 10*0.12=1.2 Variance σ2 = 10* 0.12 *(0.88) =1.056Std Deviation = √σ2 = √1.056 = 1.0276

Normal Probability Distribution

• It is a continuous probability distributionTwo values determine its shape• μ = mu = mean of

distribution• σ = sigma = standard

deviation of the distribution

Normal Probability DistributionRemember the Empirical Rule!!!

Standard Normal Distribution

-3 -2 -1 0 1 2 3

• µ = 0• σ =1• z score – tells us how

standard deviations away from the mean a value is:

z = (x - µ)/ σ

• We convert x valuesto z scores usingthe above formula or Excel! {Standardize}

190 290 390 490 590 690 790

Finding Normal ProbabilitiesSuppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6)

Finding Normal ProbabilitiesSolution to previous example….

X is normal with mean 8.0 and standard deviation 5.0, so X~N(8,5)Find P(X < 8.6) = NORMDIST(8.6,8,5,TRUE) = 0.5478

Z is std normal with mean 0 and standard deviation 1.0, so Z~N(0,1) Find P(Z < 0.12) = NORMSDIST(0.12) = 0.5478

If you want to find the value of X and Z using probabilities and you know the mean and standard deviation:

Using Excel,

For X value, =NORMINV(0.5478,8,5) = 8.6

For Z value, =NORMSINV(0.5478) = 0.12

Using Technology

• Excel Functions• BINOMDIST• NORMDIST• NORMSDIST• STANDARDIZE• NORMINV

Questions?