Probability, Statistics, and Justice Jeffrey S. Rosenthal Professor Department of Statistics University of Toronto jeff@math.toronto.edu www.probability.ca (Canadian Institute for the Administration of Justice, Annual Conference, October 10, 2013) (1/45) I'm a Professor of Statistics. A typical day's work: Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) About Me . (2/45) A typical day's work: Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) About Me . I'm a Professor of Statistics. (2/45) Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) About Me . I'm a Professor of Statistics. A typical day's work: (2/45) About Me . I'm a Professor of Statistics. A typical day's work: Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) (2/45) And then one day I wrote a successful book . (3/45) Opinion Polls . Then I was interviewed by the media about: (4/45) Then I was interviewed by the media about: Opinion Polls . (4/45) Crime statistics . (5/45) Pedestrian death counts... (6/45) Making decisions . (7/45) Game show strategies . (8/45) The Maple Leafs . To: [email protected] From: jeff@math.toronto.edu (Jeffrey Rosenthal) Date: Wed, 12 Apr 2006 16:11:08 -0400 (EDT) Subject: my calculations Hi Mike, good talking to you on the phone just now. I assumed that in each game, each team has probability 45% of getting two points, 10% of getting one point, and 45% of getting zero points. Those figures then lead to the following probabilities for Toronto to beat or tie each of the various other teams (in total points at end of season): Prob that Toronto beats Montreal = 0.17% Prob that Toronto ties Montreal = 0.30% Prob that Toronto ties Atlanta = 11.5% Prob that Toronto beats Atlanta = 30.2% Prob that Toronto ties Tampa Bay = 3.6% Prob that Toronto beats Tampa Bay = 2.1% * This gives a total probability of 5.8% (about one chance in 17) for Toronto to have a chance at the playoffs. (9/45) And lotteries . (10/45) . Lots of lotteries . (11/45) Auric Goldfinger: \Once is happenstance. Twice is coincidence. The third time it's enemy action." So what is the connection to JUSTICE? (12/45) Auric Goldfinger: \Once is happenstance. Twice is coincidence. The third time it's enemy action." So what is the connection to JUSTICE? (12/45) \Once is happenstance. Twice is coincidence. The third time it's enemy action." So what is the connection to JUSTICE? Auric Goldfinger: (12/45) So what is the connection to JUSTICE? Auric Goldfinger: \Once is happenstance. Twice is coincidence. The third time it's enemy action." (12/45) Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? (13/45) Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. (13/45) Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" (13/45) (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (13/45) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) (13/45) (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (13/45) and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, (13/45) Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. (13/45) Example: Can your friend distinguish Coke from Pepsi? Do a test! Guesses right the first time. Proof of ability? No, could be luck! What about twice in a row? three times? ten times? The p-value is the probability of such a result if it's just random. • Guess right once: p-value = 1=2 = 50%. • Guess right twice in a row: p-value = (1=2) × (1=2) = 25%. • Guess right five times in a row: multiply (\independent"): : p-value = (1=2) × (1=2) × (1=2) × (1=2) × (1=2) = 3:1%.