1. Statistics 200 Lecture #11 Tuesday, September 27, 2016
Textbook: Sections 7.1 through 7.4
Objectives:
• Introduce notions of personal probability and relative frequency
• Understand definitions of sample space, outcome, and event; identify these concepts in a simple probability experiment.
• Identify complementary events and handle probability calculations
• Identify mutually exclusive events and handle probability calculations
• Identify independent events and handle probability calculations

2. Observational Studies
Population, Sample
Census, Survey
Last Week…
Sampling 5
Confidence intervals
P-hat
Margin of error
Interpretation
Observational Studies
Retrospective
Prospective
Gathering Data 6
Randomized Experiments
Controls, placebos, blinding
matched-pair, block, repeated-measure

3. This week… 7 Probability Randomness Interpretations of probability
Relative Frequency
Personal Probability
Probability 7
Sample spaces and events
Complementary
Mutually Exclusive
Dependent / independent
Flawed intuition
Basic Rules
Complement rule
Addition rule
Multiplication rule
More probability practice

4. A random circumstance is one in which the outcome is unpredictable.
Randomness
The world full of random circumstances.
A random circumstance is one in which the outcome is unpredictable.
Examples:
Outcome of coin toss (Heads or Tails)
Which cards you are dealt in poker
Whether it rains tomorrow
What team will win the next Super Bowl.

5. More on randomness
Scientists can phrase more than you’d expect as random circumstance
Disease status / duration / symptoms
Your DNA.
Number of children per family
Eye-color
Result of a medical screening
Time it takes to walk from here to IST building
Number of people in line at Jamba Juice

6. Probability 1 A term we’ve probably all heard before
Most generally, a number between ___ and ___ assigned to a possible outcome of a random circumstance. 1
Two ways we can discuss probability:
Personal probability
Relative-frequency probability

7. The personal probability interpretation
The personal probability of an event is the degree to which a given individual believes that the event will happen.
A.k.a. subjective probability, since the personal probability can change from one person to another.
Examples:
Probability a specific job candidate will be a good fit for the company
Probability that life in the US will be better in 10 years.

8. Relative-Frequency Interpretation of Probability
many
Applies when a situation can be repeated ______ times
can not be used to determine what the outcome will be on a _________ occasion
Relative-frequency Probability of a specific outcome is defined as the ___________ of times it would occur over the long run.
proportion
single

9. Assign Probability: Relative Frequency Approach
make an _____________ about the physical world
Method 1
observe the relative frequency over __________repetitions
Method 2
assumption
many
*Won’t consider Personal Probability Assignment Method

10. Example: Relative Frequency Assignment
Make an Assumption about physical world
Population: company sells candy with the
ratio of red and blue candy shown here.
Event: pick a piece of red candy
Assumption: All candies are ______ likely to be picked
P(Event) = ____________
equally
3/10 = 0.3

11. Example: Relative Frequency Assignment
Observe the relative frequency over many trials
If we don’t know the population of candy, we can estimate the probability of red by drawing many candies.
For each trial, we draw a single candy, then replace it and mix up bag before conducting another trial.
This type of sampling is called _______________________.
Sampling with replacement

12. Example: Relative Frequency Assignment
0.3
red candy
As the number of trials increases, the _________ proportion of red candies approaches the true _________ proportion of red candies.
sample
population

13. All possible outcomes from an experiment
Sample Space:
All possible outcomes from an experiment
Experiment: Roll Two Dice 36
equally
____________ outcomes. All are ________ likely to occur

14. 1/36 Probability(snake eyes) = ________ Simple Event:
A set of a single outcome from the sample space
Example: The simple event is… observe snake eyes.
1/36
Probability(snake eyes) = ________

15. Set of one or more outcomes.
Event:
Set of one or more outcomes.
Example: The event is… observe a “2” on the first roll 6
6/36 = 1/6
___ outcomes in the event. Probability(event): ________

16. Set of one or more outcomes.
Event:
Set of one or more outcomes.
Example: The event is… observe a “2” on either roll.
What is the probability of this event?
1/36
2/36
6/36
11/36
12/36
Again: In this example it’s reasonable to assume each outcome in the sample space is equally likely.

17. Another example Flip two coins – a nickel and a quarter {(HH)}
Sample space: {(HH),(HT),(TH),(TT)}
Simple event: getting both heads _____
Event: Getting at least one head ______
Probability of at least one head = _____
{(HH)}
{(HH), (HT), (TH)}
3/4
Note: we often use capital letters to refer to events, such as A, B, C, …

18. Definitions & Probability Rules
Event: includes outcomes that are of interest
Complement: includes outcomes are not of interest
Box represents
___________
Sample space
Circle represents
___________
Event A
Everything outside of circle represents
_____________________
Complement of event A: A’ or AC

19. Definitions & Probability Rules
Previous example
of flipping two coins:
Event A = getting at least one Head.
Complement of A:
Not getting any heads:
{(HH), (HT),(TH)}
{(TT)}
Rule 1: Complement Rule
P(A) + P(Ac) = 1, If Ac represents the complement of A
So: P(Ac) = 1 – P(A)
In our example:
P(Ac)= ___________ = _____
1 – 3/4
1/4

20. Definitions & Probability Rules
Mutually exclusive
Two Events are _________________(disjoint)
if with a single observation, the two events do not have any _______ outcomes.
shared
No _________
Between the two events
overlap

21. Definitions & Probability Rules
Rule 2B: Additive Rule
P(A or B)=P(A) + P(B) if events A and B are mutually exclusive.
Continue coins example:
Event A = get only heads
Event B = get only tails
A, B are mutually exclusive
P(A or B) = __________ = _______ = _____
P(A) + P(B)
¼ + ¼

22. Definitions & Probability Rules
Rule 2A: Additive Rule (general)
P(A or B)=P(A) + P(B) – P(A and B).
A and B
Event A: get at least one head
Event B: get at least one tail A B
{(HH),(HT),(TH),(TT)}
A B
¾ + ¾ - ½ 1
P(A or B) = __________ = _______

23. Definitions & Probability Rules
Two events are independent if knowing that one will occur (or has occurred) _______ change the probability that the other occurs.
does not
Two events are dependent if knowing that one will occur (or has occurred) _______ the probability that the other occurs.
changes
Independent is not the same as mutually exclusive!

24. Definitions & Probability Rules
Rule 3B: Multiplication Rule
P(A and B) = P(A)×P(B) if Events A and B are independent.
Back to our standard example….
Event A: The nickel lands heads
Event B: The dime lands heads
{(HH),(HT),(TH),(TT)}
Independent events A B
(½) x (½)
P(A and B) = __________ = _______

25. Example
Maria wants to take French or Spanish, or both. But classes are closed, ands he must apply to enroll in a language class. She has a 60% chance of being admitted to French, a 50% chance of being admitted to Spanish, and a 20% chance of being admitted to both French and Spanish. If she applies to both French and Spanish, the probability that she will be enrolled in either French or Spanish (or possibly both) is….
French
Spanish
0.6
P(French) = ______
P(Spanish) = ________
P(French and Spanish) = ______
0.5
0.2

26. Example Yes No Clicker Question: Are these events independent?
The probability that she will be enrolled in either French or Spanish (or possibly both) is….
P(French or Spanish) = __________
= _______
= _____
P(French) +P(Spanish) – P(both)
– 0.2
0.9
Yes No
Clicker Question:
Are these events independent?

27. Example Yes No Clicker Question: Are these events mutually exclusive?
The probability that she will be enrolled in either French or Spanish (or possibly both) is….
P(French or Spanish) = __________
= _______
= _____
P(French) +P(Spanish) – P(both)
– 0.2
0.9
Yes No
Clicker Question:
Are these events mutually exclusive?

28. Summary of Rules
Rule 1: Complement Rule P(A) + P(Ac) = 1 if Ac represents the complement of A Rule 2B: Additive Rule P(A or B) = P(A) + P(B) if Events A and B are mutually exclusive Note: two events that are complements are always mutually exclusive Rule 3B: Multiplication Rule P(A and B) = P(A)×P(B) if Events A and B are independent

29. If you understand today’s lecture…
7.9, 7.11, 7.17, 7.23, 7.25, 7.29, 7.33, 7.39, 7.41, 7.43, 7.45, 7.57, 7.59
Objectives:
• Introduce notions of personal probability and relative frequency
• Understand definitions of sample space, outcome, and event; identify these concepts in a simple probability experiment.
• Identify complementary events and handle probability calculations
• Identify mutually exclusive events and handle probability calculations
• Identify independent events and handle probability calculations