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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
Citation preview
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?