Statistics: Unlocking the Power of Data Lock 5 STAT 250 Dr. 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 Lock 5 Exam 1 Friday, 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 Lock 5 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 Lock 5 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 Lock 5

Ways of Expressing Probability 1 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 % of women will get breast cancer The probability of breast cancer for a female is These statements are all saying the same thing

Statistics: Unlocking the Power of Data Lock 5 Probability statements can be directly translated into a relative frequency table: Relative Frequency Table Get Breast CancerDo not Get Breast CancerTOTAL

Statistics: Unlocking the Power of Data Lock 5 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 CancerDo not Get Breast CancerTOTAL Get Breast CancerDo not Get Breast CancerTOTAL

Statistics: Unlocking the Power of Data Lock 5 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 Lock 5 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 Lock 5 Probability to Odds

Statistics: Unlocking the Power of Data Lock 5 Odds If p = 2/3, what are the odds? a) 1 b) ½ c) 2 d) 3

Statistics: Unlocking the Power of Data Lock 5 Odds to Probability We 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 Lock 5 Odds If the odds are 1:2, what is the probability? a) 1 b) 1/3 c) 1/2 d) 2 e) 3

Statistics: Unlocking the Power of Data Lock 5 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 Lock 5 Conditional Probability The 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 Lock 5 P(survival if advanced stage) = 0.27 P(survival if early detection) = 0.98

Statistics: Unlocking the Power of Data Lock 5 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 Lock 5 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 Lock 5 What does this tell us? a) P(breast cancer if first-degree relative) = 1/16 b) P(breast cancer if first-degree relative) = 1/8 c) P(breast cancer if first-degree relative) = 1/4 d) P(breast cancer if first-degree relative) = 3/8

Statistics: Unlocking the Power of Data Lock 5 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 Lock 5 What does this tell us? a) P(family history if breast cancer) = 0.15 b) P(breast cancer if family history) = 0.15 c) P(breast cancer if no family history) = 0.15

Statistics: Unlocking the Power of Data Lock 5

1 in 8 women (12.5%) of women get breast cancer, so P(breast cancer if female) = in 800 (0.125%) of men get breast cancer, so P(breast cancer if male) =

Statistics: Unlocking the Power of Data Lock 5 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 Lock 5 Odds Ratio The 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 Lock 5 Odds Ratio Odds ratio for breast cancer comparing females to males:

Statistics: Unlocking the Power of Data Lock 5 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) % Breast Cancer Screening

Statistics: Unlocking the Power of Data Lock 5 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 Lock 5 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) % Breast Cancer Screening

Statistics: Unlocking the Power of Data Lock 5 Breast Cancer Screening A 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 Lock 5 CancerCancer-free Positive Result Negative 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. Everyone We randomly pick a ball from the Everyone bin. C C C C C FFFFFFFFFFFF FFFFFFFFFFF FFFFFFFFFF FFFFFFFFF FFFFFFFF FFFFFFF FFFFFF FFFFF 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 Lock 5 100,000 women in the population 1% 1000 have cancer99,000 cancer-free 99% 80%20% 800 test positive 200 test negative 9.6%90.4% 9,504 test positive 89,496 test negative Thus, 800/(800+9,504) = 7.8% of positive results have cancer

Statistics: Unlocking the Power of Data Lock 5 Two-Way Table Create 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 Lock 5 Mammograms 1. Probability 1. What’s the probability of breast cancer if a positive mammogram? 2. What’s the probability of breast cancer if a negative mammogram? 2. Odds 1. 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.

Statistics: Unlocking the Power of Data Lock 5

To Do Read Section 2.1 Do HW 2.1, 11 (due Friday, 2/13) Study for Exam 1 (Friday, 2/13)