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# Psychology, Science, and Pseudoscience: Class #06 (Probability Fails)

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In short: Even extremely well educated and qualified people have great difficulty dealing with (a) randomness or (b) probabilistic reasoning

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### Psychology, Science, and Pseudoscience: Class #06 (Probability Fails)

1. 1. PS409 Psychology, Science, & Pseudoscience Dr Brian Hughes School of Psychologybrian.hughes@nuigalway.ie @b_m_hughes
2. 2. Evidentiary reasoning:Why do people believe weird things?
3. 3. Difficulties with probabilistic reasoningTossing one coin, what is the chanceof it landing on “Heads”?A: 0.50 (or, in other words, a fifty-fifty chance)CHECK HERE Throwing one dice, what is the chance of it landing on “5”? A: 0.17 (or, in other words, a one-in-six chance)
4. 4. Difficulties with probabilistic reasoningTossing two coins, what is thechance of getting two “Heads”?A: 0.25 (or, in other words, a 1-in-4 chance)CHECK HERE Throwing two dice, what is the chance of getting two “5”s? A: 0.028 (or, in other words, a 1-in-36 chance)
5. 5. Difficulties with probabilistic reasoningImagine you are at a party, where there are 23 peoplepresent (including yourself). What are the chances thattwo of these 23 people share the same birthday?(a) 1 chance in 365, or 1/365(b) Around 1/1,000(c) Around 1/2(d) Around 1/40(e) 1/2,020 Paulos (1988)
6. 6. Difficulties with probabilistic reasoningImagine you are at a party, where there are 23 peoplepresent (including yourself). What are the chances thattwo of these 23 people share the same birthday?(a) 1 chance in 365, or 1/365(b) Around 1/1,000(c) Around 1/2(d) Around 1/40(e) 1/2,020 Paulos (1988)
7. 7. Difficulties with probabilistic reasoningIf a test to detect a disease whose prevalence is 1/1000 has a falsepositive rate of 5%, what is the chance that a person found to have apositive result actually has the disease, assuming you know nothingabout the person’s symptoms or signs? • Among Staff and Students of Harvard Medical School (n = 60): Most popular answer = 0.95 (or, in other words, a 19 out of 20 chance) Average of all answers = 0.56 (or, in other words, around a fifty-fifty chance) Cited by Pinker (1997)
8. 8. Difficulties with probabilistic reasoningIf a test to detect a disease whose prevalence is 1/1000 has a falsepositive rate of 5%, what is the chance that a person found to have apositive result actually has the disease, assuming you know nothingabout the person’s symptoms or signs? [Base-rate] x [Test sensitivity] / [Rate of positive results] Prevalence of disease Proportion of sick who Number of positive per 1000 test positive results per 1000 Actual sick the “false” [1/1000] [1/1] [ persons & positives ] Actual sick “False positives” persons testing i.e., well persons who ‘positive’ test ‘positive’ Cited by Pinker (1997)
9. 9. Difficulties with probabilistic reasoningIf a test to detect a disease whose prevalence is 1/1000 has a falsepositive rate of 5%, what is the chance that a person found to have apositive result actually has the disease, assuming you know nothingabout the person’s symptoms or signs? [Base-rate] x [Test sensitivity] / [Rate of positive results] Prevalence of disease Proportion of sick who Number of positive per 1000 test positive results per 1000 [1/1000] [1/1] [ 1/1000 + ( 999/1000 x .05 )] Actual sick “False positives” persons testing i.e., well persons who ‘positive’ test ‘positive’ Cited by Pinker (1997)
10. 10. Difficulties with probabilistic reasoningIf a test to detect a disease whose prevalence is 1/1000 has a falsepositive rate of 5%, what is the chance that a person found to have apositive result actually has the disease, assuming you know nothingabout the person’s symptoms or signs? [Base-rate] x [Test sensitivity] / [Rate of positive results]= 0.001 x 1.0 / [ 0.001 + ( 0.04995 )]= 0.001 / 0.05095= 0.019627≈ 0.02 Cited by Pinker (1997)
11. 11. Difficulties with probabilistic reasoningIf a test to detect a disease whose prevalence is 1/1000 has a falsepositive rate of 5%, what is the chance that a person found to have apositive result actually has the disease, assuming you know nothingabout the person’s symptoms or signs?≈ 0.02 (or, in other words, a 1-in-50 chance) Cited by Pinker (1997)
12. 12. Difficulties with probabilistic reasoningIf a test to detect a disease whose prevalence is 1/1000 has a falsepositive rate of 5%, what is the chance that a person found to have apositive result actually has the disease, assuming you know nothingabout the person’s symptoms or signs?≈ 0.02 (or, in other words, a 1-in-50 chance) • Among Staff and Students of Harvard Medical School (n = 60): Most popular answer = 0.95 (or, in other words, a 19 out of 20 chance) Average of all answers = 0.56 (or, in other words, around a fifty-fifty chance) Cited by Pinker (1997)
13. 13. Difficulties with probabilistic reasoning Gøtzsche PC, Nielsen M. Screening for breast cancer with mammography. Cochrane Database of Systematic Reviews 2006, Issue 4. Art. No.: CD001877. DOI: 10.1002/14651858.CD001877.pub2
14. 14. Difficulties with probabilistic reasoning
15. 15. Difficulties with probabilistic reasoning
16. 16. Difficulties with probabilistic reasoning
17. 17. Difficulties with probabilistic reasoning Lucia de Berk (Wikipedia) Dutch nurse sentenced to life imprisonment in 2003 Found guilty of four murders and three attempted murders, largely on statistical evidence “one in 342 million against” Problems: Multiplied p-values, as per coin tosses Did not compare against base-rate Case re-opened in 2008, for 2009 hearing Exonerated in April 2010
18. 18. Difficulties with probabilistic reasoning Sally Clark (Wikipedia) British lawyer convicted in 1999 of murdering her two babies Professor Roy Meadow, claimed odds of two deaths were “one in 73 million” Problems: Multiplied p-values, as per coin tosses; actual odds are 1 in 10,000 Did not compare against base rate: Which is rarer, double SIDS or double murder? Clark released in 2003, died in 2007
19. 19. Probability of coincidencesWhat is the probability that you wereborn in the same month as BarackObama? August 1 in 12 (or 0.08)What is the probability that you wereborn on the same day-of-the-week asBarack Obama? Friday 1 in 7 (or 0.14)What is the probability that you share Born 1961your birthday with Barack Obama? Note: The probability of any August 4th date-based coincidence is 1 in 365.25 (or 0.003) 1 in 365.25
20. 20. Probability of coincidencesWhat are the chances of an “uncanny coincidence”? Imagine 100 trivial events per day. This produces 4,950 possible pairings or coincidences (99 + 98 + 97…). For 1,000 people across 10 years, such a rate produces 18,067,500,000 pairings. In Galway, we would have 222,750,000 pairings every day (or 81,303,750,000 per year) At least some of these pairings will be uncanny! Marks & Kammann (1980)
21. 21. PS409 Psychology, Science, & Pseudoscience Dr Brian Hughes School of Psychologybrian.hughes@nuigalway.ie @b_m_hughes