Wednesday, January 12 Topic: Probability and Randomness Read: Utts 15 Read: Utts 16 Vocab: probability, relative frequency, personal probability, coherent, Vocab: mutually exclusive events, independents events, Vocab: cummulative probability, expected value, four probability rules Due: HW #4 @ hw02.shtml Review Chi-Squared Hugo -- 5 times in 12 rolls; how unusual is that? have students roll 12 dice several times and count number of 6's rolled (work in pairs) Berkeley admissions (1=men 2=women) (Utts page 221, exercise 14) video clip -- FAPP #10 2:04:45 -- 2:08:40 [didn't show] Expected counts are printed below observed counts accept reject Total 1 450 550 1000 416.67 583.33 2 175 325 500 208.33 291.67 Total 625 875 1500 Chi-Sq = 2.667 + 1.905 + 5.333 + 3.810 = 13.714 DF = 1, P-Value = 0.000 ------------------------------------------------ Simpson's Parodox If we divide by programs applied to, we see a different story ------------------------------------------------ Expected counts are printed below observed counts acceptA rejectA Total 1 400 250 650 403.45 246.55 2 50 25 75 46.55 28.45 Total 450 275 725 Chi-Sq = 0.029 + 0.048 + 0.255 + 0.418 = 0.751 DF = 1, P-Value = 0.386 ------------------------------------------------ Expected counts are printed below observed counts acceptB rejectB Total 1 50 300 350 79.03 270.97 2 125 300 425 95.97 329.03 Total 175 600 775 Chi-Sq = 10.665 + 3.111 + 8.783 + 2.562 = 25.120 DF = 1, P-Value = 0.000 ------------------------------------------------ hospital example (Utts chapter 12, pages 213-215) give combined results first, then separate survive die s rate d rate standard 505 595 .46 .54 new 195 905 .18 .82 total 700 1500 standard 5 95 .05 .95 new 100 900 .10 .90 total 105 995 discrimination example (Utts, chapter 12, pages 215-217) ?? death penalty 326 cases, white defendant: 19/160 get death pen. (.119) black defendant: 17/166 get death pen. (.102) when separated by victim's race, see different story [overhead from Moore 207] point: statistically significant means that the effect is not likely to be due to chance alone, but there may be some other factor than the obvious one that is reason Probability random: long-term predictability vs short-term unpredictability law of large numbers / "law" of small numbers scale: 0 to 1 (0% to 100%) personal vs. mathematical (relative frequency) 4 Rules and applications axiomatic method four rules "overhead" examples probability of losing luggage is 1/176 (Krantz) P(heart attack kills) = .33, P(cancer kills) = .2 [assuming death] estimated probability of grades probability of two girls (P(boy) about .512) probability of winning 2 of 3, 3 of 5 given an estimate for each game Expected Value lottery example insurance video -- Life By the Numbers (#4 Prob) 02:00 (or 08:20) - 27:00: intro to prob., Graunt, casinos 27:00 - 42:25: polling, polio, prob assesses results