1. Unit 6: Probability

2. Expected values Ex 1: Flip a coin 10 times, paying $1 to play each time. You win $.50 (plus your $1) if you get a head. How much should you expect to win? Ex 2: Roll two dodecahedral (12-sided) dice. You win $10 (plus your payment to play) if you get doubles. How much should you pay to play for a fair game?

3. Two similar examples: From text: Paradox of the Chevalier de la Méré: P(at least 1 ace in 4 rolls of die) > P(at least 1 double-ace in 24 rolls of 2 dice) Birthday problem: With 30 people in a room, how likely is it that at least two have the same birth date?

4. The Birthday problem # peopleP(no match)P(match) 20.997260270.00273973 30.991795830.00820417 40.983644090.01635591 50.972864430.02713557 60.959537520.04046248 70.94376430.0562357 80.925664710.07433529 90.905376170.09462383 100.883051820.11694818 110.858858620.14114138 120.832975210.16702479 130.805589720.19441028 140.776897490.22310251 150.747098680.25290132 160.716395990.28360401 170.684992330.31500767 180.653088580.34691142 190.620881470.37911853 200.588561620.41143838 210.556311660.44368834 220.524304690.47569531 230.492702770.50729723 240.461655740.53834426 250.43130030.5686997 260.401759180.59824082 270.373140720.62685928 280.345538530.65446147 290.319031460.68096854 300.293683760.70631624 310.269545370.73045463 320.246652470.75334753 330.225028150.77497185 340.204683140.79531686

5. Tree diagram Flip a coin, then roll a die, list all alternatives

6. The Monty Hall Problem (From Marilyn vos Savant’s column) Game show: Three doors hide a car and 2 goats. Contestant picks a door. Host opens one of the other doors to reveal a goat. Contestant then may switch to the other unopened door. Is it better to stay with the original choice or to switch; or doesn’t it matter? Marilyn’s answer: Switch! Many respondents: Doesn’t matter. (“You’re the goat!”)

7. Tree diagram of Stayer’s possible games

9. Math stuff about binomial coeffients They’re called that because they are the coefficients of x and y in the expansion of (x+y) n : –C(n,0)x n + C(n,1)x n-1 y + C(n,2)x n-2 y 2 +... + C(n,n-1)xy n-1 + C(n,n)y n For small n, compute C(n,k) with “Pascal’s triangle”: 1’s in first row and column, then each entry is sum of the one above and the one to the right

10. (More from Marilyn vos Savant’s column) Suppose we assume that 5% of the people are drug users. A drug test is 95% accurate (i.e., it produces a correct result 95% of the time, whether the person is using drugs or not). A randomly chosen person tests positive. Is the person highly to be a drug user? Marilyn’s answer: Given your conditions, once the person has tested positive, you may as well flip a coin to determine whether she or he is a drug user. The chances are only 50- 50. But the assumptions, the make-up of the test group and the true accuracy of the tests themselves are additional considerations. (To see this, suppose the population is 10,000 people; compare numbers of false positives and true positives.)

11. Drug [disease] testing probabilities Drug [disease] present?Test positiveTest negative Yes“Sensitivity”False negative NoFalse positive“Specificity” Ex: Suppose the Bovine test for lactose abuse has a sensitivity of 0.99 and a specificity of 0.95; and that 7% of a certain population abuses lactose. If a person tests positive on the Bovine test, how likely is it that (s)he really abuses lactose? Sum 1 posneg abuser.99.01 clean.05.95

12. Assuming 7% of population is really positive: x = sensitivity y = specificity z = P(pos test => pos) curve: x =.99 points: x =.99 y =.95,.90 z =.6,.42

13. Counting dragonflies (thanks to Profs. V. MacMillen and R. Arnold)

16. Only two pairs 30 censuses altogether, 17 with only two pairs Of 17, 12 had both in same plot Do they prefer to lay eggs in proximity?

17. Censuses with >2 pairs P1P2P3 003 030 300 102 120 300 120 210 120 003 120 P1P2P3 400 004 310 400 040 022 P1P2P3 320 005 032 311 230 230 131 104 122 014 005 Up to 12 at the same time

18. With 3 pairs