Transcript PowerPoint

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
Last Week…
5
6
Population, Sample
Census,
Survey
Confidence intervals
• P-hat
• Margin of error
• Interpretation
Sampling
Gathering Data
Observational Studies
Retrospective
Prospective
Randomized Experiments
Controls, placebos, blinding
matched-pair, block, repeated-measure
This week…
Randomness
Interpretations of probability
7 Probability
• Relative Frequency
• Personal Probability
Sample spaces and events
• Complementary
• Mutually Exclusive
• Dependent / independent
Flawed intuition
More probability
practice
Basic Rules
• Complement rule
• Addition rule
• Multiplication rule
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.
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
Probability
A term we’ve probably all heard before
0 and
Most generally, a number between ___
1
___
assigned to a possible outcome of a
random circumstance.
Two ways we can discuss probability:
1. Personal probability
2. Relative-frequency probability
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
• Applies when a situation can be repeated
many times
______
Relative-frequency Probability of a specific
proportion
outcome is defined as the ___________
of
times it would occur over the long run.
• can not be used to determine what the outcome will be on a
single occasion
_________
9
Assign Probability:
Relative Frequency Approach
Method 1
assumption
• make an _____________
about the physical world
Method 2
• observe the relative
frequency over
__________repetitions
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
equally likely to be picked
Assumption: All candies are ______
3/10 = 0.3
P(Event) = ____________
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
sample
As the number of trials increases, the _________
proportion
population proportion of
of red candies approaches the true _________
red candies.
13
Sample Space:
All possible outcomes from an experiment
Experiment: Roll Two Dice
36
equally likely to occur
____________
outcomes. All are ________
14
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
Event:
Set of one or more outcomes.
Example: The event is… observe a “2” on the first roll
6 outcomes in the event. Probability(event): ________
6/36 = 1/6
___
16
Event:
Set of one or more outcomes.
Example: The event is… observe a “2” on either roll.
What is the probability of this event?
(A) 1/36
(B) 2/36
(C) 6/36
(D) 11/36
(E) 12/36
Again: In this example it’s reasonable to assume
each outcome in the sample space is equally likely.
Another example
Flip two coins – a nickel and a quarter
Sample space: {(HH),(HT),(TH),(TT)}
{(HH)}
Simple event: getting both heads _____
{(HH), (HT), (TH)}
Event: Getting at least one head ______
Probability of at least one head =
3/4
_____
Note: we often use capital
letters to refer to events,
such as A, B, C, …
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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
C
Complement
of
event
A:
A’
or
A
_____________________
Definitions & Probability Rules
Previous example
of flipping two coins:
Event A = getting at least one
Head.
{(HH), (HT),(TH)}
Complement of A:
Not getting any heads: {(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:
1 – 3/4
1/4
P(Ac)= ___________
= _____
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Definitions & Probability Rules
• Two Events are
Mutually exclusive (disjoint)
_________________
if with a single observation, the two events do not have
shared outcomes.
any _______
overlap
No _________
Between the two events
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
½
P(A) + P(B) = _______
¼ + ¼ = _____
P(A or B) = __________
Definitions & Probability Rules
Rule 2A: Additive Rule (general)
P(A or B)=P(A) + P(B) – P(A and B).
A and B
A
B
Event A: get at least one head
Event B: get at least one tail
{(HH),(HT),(TH),(TT)}
A
¾ + ¾ - ½ = _______
1
P(A or B) = __________
B
Definitions & Probability Rules
Two events are independent if knowing that one
does not change the
will occur (or has occurred) _______
probability that the other occurs.
Two events are dependent if knowing that one will
changes the probability
occur (or has occurred) _______
that the other occurs.
Independent is not the same as mutually exclusive!
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
Independent events
{(HH),(HT),(TH),(TT)}
A
B
(½) x (½) = _______
P(A and B) = __________
¼
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
0.6
P(French) = ______
0.5
P(Spanish) = ________
0.2
P(French and Spanish) = ______
Spanish
Example
The probability that she will be enrolled in either French or
Spanish (or possibly both) is….
P(French) +P(Spanish) – P(both)
P(French or Spanish) = __________
0.6 + 0.5 – 0.2
= _______
0.9
= _____
Clicker Question:
Are these events independent?
A. Yes
B. No
Example
The probability that she will be enrolled in either French or
Spanish (or possibly both) is….
P(French) +P(Spanish) – P(both)
P(French or Spanish) = __________
0.6 + 0.5 – 0.2
= _______
0.9
= _____
Clicker Question:
Are these events mutually
exclusive?
A. Yes
B. No
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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
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