Mar. 3 Statistic for the day: Average number of pieces of mail that end up in the dead letter box each year: 57,100,000 Assignment: Read Chapter 17 Exercises p : 1, 4, 7, 10, 15 These slides were created by Tom Hettmansperger and in some cases modified by David Hunter

Shuffle two decks of cards. Stack the two decks side-by-side, face down next to each other. Stack the two decks side-by-side, face down next to each other. One by one, flip over one card from each deck. One by one, flip over one card from each deck. I bet I see at least one match. Do you want to bet against me? I bet I see at least one match. Do you want to bet against me?

Probability of no match: Probability of no match on 1 st flip: 51/52 Probability of no match on 1 st flip: 51/52 Probability of no match on 2 nd flip: 51/52 Probability of no match on 2 nd flip: 51/52 … Probability of no match on 52 nd flip: 51/52 Probability of no match on 52 nd flip: 51/52 These events are NOT independent; however, they are APPROXIMATELY independent because, say, whether a match occurs on the 36th flip doesn’t influence whether a match occurs on the 47th flip very strongly.

Efron Dice D C B A Side value

Die B Die A

Die B Pr(B beats A) = ( )/36 = 24/36 = 2/3

Die B Die C Pr(C beats B) = (18 + 6)/36 = 24/36 = 2/3

Die D Die C Pr(D beats C) = 24/36 = 2/3

Die A Die D Pr( A beats D ) = 24/36 = 2/3

333333D C B A Side value Pr( B beats A ) = 2/3 Pr( C beats B ) = 2/3 Pr( D beats C ) = 2/3 Pr (A beats D ) = 2/3 Hence, there is NO best die! You can always pick a winner if you pick second.

Percent tables and count tables A stratified population is one that is divided into mutually exclusive subgroups and the subgroups exhaust all members of the population.

Cancer testing: confusion of the inverse Suppose we have a cancer test for a certain type of cancer. Sensitivity of the test: If you have cancer then the probability of a positive test is.98. Pr(+ given you have C) =.98 Specificity of the test: If you do not have cancer then the probability of a negative test is.95. Pr(- given you do not have C) =.95 Base rate: The percent of the population who has the cancer. This is the probability that someone has C. Suppose for our example it is 1%. Hence, Pr(C) =.01.

+Positive-Negative C(Cancer) no C (no Cancer) Sensitivity Specificity Base Rate Percent table Suppose you go in for a test and it comes back positive. What is the probability that you have cancer? false positivefalse negative

Count table from a percent table+- C no C C , ,000 Pr(C given a + test) = 98/593 =.165

Do you have a tattoo? What is the probability that a randomly chosen person from the class will say yes? Rows: Sex Columns: Tattoo No Yes All Female Male All Need a count table to estimate the probabilities:

Rows: Sex Columns: Tattoo No Yes All Female Male All Percent table: Pr(yes) = 46/236 =.1949 Pr(yes given the person is a female) =.2279 Pr(yes given the person is a male) =.1500 Are the events ‘yes’ and ‘female’ independent?

Pr(no given the person is female) =.7721 Pr(no given the person is a male) =.8500 Suppose I tell you that a stat100 student came into office hours and they said that they did not have a tattoo. Which is more likely: The student was female. The student was male.

Rows: Sex Columns: Tattoo No Yes All Female Male All Pr(female given the student said no) = 105/190 =.553 Pr(male given the student said no) = 85/190 =.447 More likely that the student is a female!

Rows: Sex Columns: Tattoo No Yes All Female Male All Pr(yes) = 46/236 =.195 Pr(no) = 190/236 =.805 Pr(female) = 136/236 =.576 Pr(male) = 100/236 =.424 Pr(yes given the student is a female) = 31/136 =.228 Pr(yes given the student is a male) = 15/100 =.150 Pr(no given the student is a female) = 105/136 =.772 Pr(no given the student is a male) = 85/100 =.850 Pr(female given the student said yes) = 31/46 =.674 Pr(male given the student said yes) = 15/46 =.326 Pr(female given the student said no) = 105/190 =.553 Pr(male given the student said no) = 85/190 =.447

The count table gives the ability to calculate everything. If you have a percent table, you should create a count table. Rows: Sex Columns: Tattoo No Yes All Female Male All Note: It’s not always possible to reconstruct a representative count table. In the above, you can’t do it unless you also know the percentage of females.

The count table gives the ability to calculate everything. If you have a percent table, you should create a count table. Rows: Sex Columns: Tattoo No Yes All Female Male All Also, 57.63% are females NY F M (arbitrary) Leads to: