1. Welcome to MM305 Unit 3 Seminar Dr
Welcome to MM305 Unit 3 Seminar Dr. Bob Probability Concepts and Applications

2. The Basics of Probability
Events
Outcomes
Probability Experiment
Sample Space

3. 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

4. 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

5. 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

6. 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

7. 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

8. Probability: Tying it all together
Blood Alcohol Level of Victim
0.00%
(A) %
(B)
≥0.10%
(C)
Total
0-19
(D)
142 7 6
155
20-39
(E) 47 8 41 96
40-49
(F) 29 77
114
Over 60
(G) 35 89
265 30
159
454
Age

9. Venn Diagrams P (A) P (B) P (A) P (B) P (A and 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)

10. 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.

11. 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

12. Using Technology—Excel Template

13. Binomial Distribution
1: 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).

14. 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

15. 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

16. 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

17. Excel Function: BINOMDIST
P(X=3) = BINOMDIST(3,10,0.12,FALSE) =
P(X<=3) = BINOMDIST(3,10,0.12,TRUE) =
P(X>3) = 1 - P(X<=3) = =
E(X)= n*p= 10*0.12= Variance σ2 = 10* 0.12 *(0.88) =1.056
Std Deviation = √σ2 = √ =

18. 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

19. Normal Probability Distribution
Remember the Empirical Rule!!!

20. 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}

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

22. 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) =
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) =
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

23. Using Technology Excel Functions BINOMDIST NORMDIST NORMSDIST
STANDARDIZE
NORMINV

24. Questions?