1. 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. 309-311: 1, 4, 7, 10, 15 These slides were created by Tom Hettmansperger and in some cases modified by David Hunter

2. 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?

3. 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.

4. Efron Dice 333333D 662222C 555111B 444400A Side value

5. 4 4 4 4 0 0 555111 Die B Die A

6. 4 4 4 4 0 0 555111 Die B Pr(B beats A) = (12 + 12)/36 = 24/36 = 2/3

7. 5 5 5 1 1 1 662222 Die B Die C Pr(C beats B) = (18 + 6)/36 = 24/36 = 2/3

8. 6 6 2 2 2 2 333333 Die D Die C Pr(D beats C) = 24/36 = 2/3

9. 3 3 3 3 3 3 444400 Die A Die D Pr( A beats D ) = 24/36 = 2/3

10. 333333D 662222C 555111B 444400A 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.

11. 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.

12. 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.

13. +Positive-Negative C(Cancer).98.02.01 no C (no Cancer).05.95.99 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

14. Count table from a percent table+- C.98.02.01 no C.05.95.99 +-C982100 49594059,900 593940710,000 Pr(C given a + test) = 98/593 =.165

15. 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 105 31 136 Male 85 15 100 All 190 46 236 Need a count table to estimate the probabilities:

16. Rows: Sex Columns: Tattoo No Yes All Female 77.21 22.79 100.00 Male 85.00 15.00 100.00 All 80.51 19.49 100.00 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?

17. 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.

18. Rows: Sex Columns: Tattoo No Yes All Female 105 31 136 Male 85 15 100 All 190 46 236 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!

19. Rows: Sex Columns: Tattoo No Yes All Female 105 31 136 Male 85 15 100 All 190 46 236 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

20. 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 77.21 22.79 100.00 Male 85.00 15.00 100.00 All 80.51 19.49 100.00 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.

21. 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 77.21 22.79 100.00 Male 85.00 15.00 100.00 All 80.51 19.49 100.00 Also, 57.63% are females NY F445013135763 M63636014237 10000 (arbitrary) Leads to: