Transcript Document

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 Probability
Three 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 Probability
Independent Events
Two 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 Events
Events 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
Blood Alcohol Level of Victim
0.00%
(A)
0.01-0.09%
(B)
≥0.10%
(C)
Total
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
265
30
159
454
Age
0-19
(D)
Total
Venn Diagrams
P (A and B)
P (A)
P (B)
P (A)
P (B)
Events that are
mutually exclusive
Events that are not
mutually exclusive
P (A or B) = P (A) + P (B)
P (A or B) =
P (A) + P (B) – P (A and B)
Random Variables
A random variable assigns a real number to every
possible outcome or event in an experiment
Discrete random variables can assume only a finite
or limited set of values
Continuous random variables can assume any one
of an infinite set of values
Always define what your random variable represents!
Let X = number of people, companies, computers, hours, etc.
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 Distribution
1:
2:
3:
4:
The number of trials n is fixed.
Each trial is independent.
Each trial represents one of two outcomes ("success" or "failure").
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 Distribution
Each trial has only two possible outcomes
 The probability stays the same from one trial
to the next
The trials are statistically independent
The 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 Distribution
Suppose 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=3
p=0.12
q=1-0.12=0.88
Excel Function: BINOMDIST
P(X=3) = BINOMDIST(3,10,0.12,FALSE) = 0.0847
P(X<=3) = BINOMDIST(3,10,0.12,TRUE) = 0.9761
P(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) =1.056
Std Deviation = √σ2 = √1.056 = 1.0276
Normal Probability Distribution
• It is a continuous
probability distribution
Two values determine its
shape
• μ = mu = mean of
distribution
• σ = sigma = standard
deviation of the
distribution
Normal Probability Distribution
Remember the Empirical Rule!!!
Standard Normal Distribution
•
•
•
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 values
to z scores using
the above formula
or Excel! {Standardize}
-3
190
-2
290
-1
390
0
490
1
590
2
690
3
790
Finding Normal Probabilities
Suppose X is normal with mean 8.0 and standard deviation 5.0.
Find P(X < 8.6)
Finding Normal Probabilities
Solution 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?