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Statistics: Unlocking the Power of Data Lock5
STAT 250Dr. Kari Lock Morgan
Probability
SECTIONS 11.1• Probability (11.1)• Odds, odds ratio (not in book)• Conditional probability (11.1)
Statistics: Unlocking the Power of Data Lock5
Exam 1Friday, 2/13, in class
You can bring One single-sided page (8 ½ x 11) of notes A non-cell phone calculator
Exam will cover Everything covered through today’s class Chapters 1 and 2, plus today’s material Lecture and lab (but not Minitab commands)
Mix of multiple choice and free response
Statistics: Unlocking the Power of Data Lock5
Event• An event is something that either happens or doesn’t happen, or something that either is true or is not true
• Examples:• You get cancer• A randomly selected person is obese• A particular mutation occurs• It snows tonight
Statistics: Unlocking the Power of Data Lock5
Probability• The probability of event A, P(A), is the probability that A will happen
• Probability always refers to an event
• Probability is always between 0 and 1
• P(A) = 1 means A will definitely happen
• P(A) = 0 means A will definitely not happen
Statistics: Unlocking the Power of Data Lock5
Ways of Expressing Probability1 in 8 women will get breast cancer
1/8 of women will get breast cancer
The proportion of women who will get breast cancer is 1/8, or 0.125.
12.5% of women will get breast cancer
The probability of breast cancer for a female is 0.125
These statements are all saying the same thing
Statistics: Unlocking the Power of Data Lock5
Probability statements can be directly translated into a relative frequency table:
Relative Frequency Table
Get Breast Cancer Do not Get Breast Cancer TOTAL
Statistics: Unlocking the Power of Data Lock5
Probability statements can also be translated into a frequency table, although unlike data description, the total is arbitrary:
Any of these are equally valid for probability calculations!
Frequency Table
Get Breast Cancer Do not Get Breast Cancer TOTAL
Get Breast Cancer Do not Get Breast Cancer TOTAL
Statistics: Unlocking the Power of Data Lock5
Odds
If p denotes the probability of an event, the odds are defined as
Interpreting odds Odds of 1 indicate 50/50 p < 0.5 yield odds < 1 p > 0.5 yield odds > 1
Odds of 3, or 3:1, mean that out of 4 times, we would expect the variable to be in that category 3 times and out of that category 1 time
Statistics: Unlocking the Power of Data Lock5
Breast Cancer Odds
Commonly expressed as 1:7 or 1/7
The odds of a woman getting breast cancer are 1:7
For every one woman who will get breast cancer, 7 women will not.
Statistics: Unlocking the Power of Data Lock5
Odds to ProbabilityWe can go from odds to probability with
One may make more sense than the other to you, so be comfortable going back and forth
Statistics: Unlocking the Power of Data Lock5
Odds
If the odds are 1:2, what is the probability?
a) 1b) 1/3c) 1/2d) 2e) 3
Statistics: Unlocking the Power of Data Lock5
Conditional Probability• P(A if B) is the probability of A, if we know B has happened or is true
• This is read in multiple ways:
• “probability of A if B”• “probability of A given B” • “probability of A conditional on B”
• You may also see this written as P(A | B)
Statistics: Unlocking the Power of Data Lock5
Conditional ProbabilityThe probability and odds that we calculated are
restricted only to females, so we are implicitly conditioning on the fact that gender is female.
For females, what’s the chance of getting breast cancer?
What proportion of women will get breast cancer?
Conditional probability is the probability of an event, conditional on (or given) that another variable takes a specific value (gender = female)
Statistics: Unlocking the Power of Data Lock5
P(survival if advanced stage) = 0.27P(survival if early detection) = 0.98
Statistics: Unlocking the Power of Data Lock5
What does this tell us?
a) P(breast cancer if 30 – 39 years old)
b) P(30 – 39 years old if breast cancer)
Statistics: Unlocking the Power of Data Lock5
What does this tell us?
a) P(breast cancer if first-degree relative)
b) P(first degree relative if breast cancer)
Statistics: Unlocking the Power of Data Lock5
What does this tell us? a) P(breast cancer if first-degree relative) = 1/16b) P(breast cancer if first-degree relative) = 1/8c) P(breast cancer if first-degree relative) = 1/4d) P(breast cancer if first-degree relative) = 3/8
Statistics: Unlocking the Power of Data Lock5
What does this tell us?
a) P(breast cancer if no family history)
b) P(no family history if breast cancer)
Statistics: Unlocking the Power of Data Lock5
What does this tell us? a) P(family history if breast cancer) = 0.15b) P(breast cancer if family history) = 0.15c) P(breast cancer if no family history) = 0.15
Statistics: Unlocking the Power of Data Lock5
1 in 8 women (12.5%) of women get breast cancer, so P(breast cancer if female) = 0.125
1 in 800 (0.125%) of men get breast cancer, so P(breast cancer if male) = 0.00125
Statistics: Unlocking the Power of Data Lock5
Two-Way Table
What’s the overall (unconditional) probability of breast cancer?
Create a two-way table, with 1000 each of males and females.
Statistics: Unlocking the Power of Data Lock5
Odds RatioThe odds ratio (OR) is the ratio of the odds of
an event in one group to the odds of an event in another group
Odds ratio for breast cancer comparing females to males:
Statistics: Unlocking the Power of Data Lock5
Odds RatioOdds ratio for breast cancer comparing
females to males:
Statistics: Unlocking the Power of Data Lock5
A 40-year old woman participates in routine screening and has a positive mammography. What’s the probability she has cancer?
a) 0-10%
b) 10-25%
c) 25-50%
d) 50-75%
e) 75-100%
Breast Cancer Screening
Statistics: Unlocking the Power of Data Lock5
1% of women at age 40 who participate in routine screening have breast cancer.
80% of women with breast cancer get positive mammographies.
9.6% of women without breast cancer get positive mammographies.
A 40-year old woman participates in routine screening and has a positive mammography. What’s the probability she has cancer?
Breast Cancer Screening
Statistics: Unlocking the Power of Data Lock5
A 40-year old woman participates in routine screening and has a positive mammography. What’s the probability she has cancer?
a) 0-10%
b) 10-25%
c) 25-50%
d) 50-75%
e) 75-100%
Breast Cancer Screening
Statistics: Unlocking the Power of Data Lock5
Breast Cancer ScreeningA 40-year old woman participates in routine screening and has a positive mammography. What’s the probability she has cancer?
What is this asking for?
a) P(cancer if positive mammography)b) P(positive mammography if cancer)c) P(positive mammography if no cancer)d) P(positive mammography)e) P(cancer)
Statistics: Unlocking the Power of Data Lock5
Cancer
Cancer-free
Positive ResultNegative Result
If we randomly pick a ball from the Cancer bin, it’s more likely to be red/positive.
If we randomly pick a ball the Cancer-free bin, it’s more likely to be green/negative.
EveryoneWe randomly pick a ball from the Everyone bin.
C
C
C
C
C
F F F F F F F F F F F FF F F F F F F F F F FF F F F F F F F F FF F F F F F F F FF F F F F F F FF F F F F F FF F F F F FF F F F F
If the ball is red/positive, is it more likely to be from the Cancer or Cancer-free bin?
Statistics: Unlocking the Power of Data Lock5
100,000 women in the population
1%
1000 have cancer 99,000 cancer-free
99%
80% 20%
800 testpositive
200 testnegative
9.6% 90.4%
9,504 testpositive
89,496 testnegative
Thus, 800/(800+9,504) = 7.8% of positive results have cancer
Statistics: Unlocking the Power of Data Lock5
Two-Way TableCreate a two-way table for mammogram
result and breast cancer status, using 10,000 women total.
Using the table, what’s P(cancer if positive)?
Statistics: Unlocking the Power of Data Lock5
Mammograms
1. Probability1. What’s the probability of breast cancer if a positive
mammogram?2. What’s the probability of breast cancer if a negative
mammogram?
2. Odds1. What’s the odds of breast cancer after a positive
mammogram?2. What’s the odds of breast cancer after a negative
mammogram?
3. Compute the odds ratio comparing breast cancer risk for positive and negative mammograms.