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PS409
   Psychology, Science,
     & Pseudoscience



           Dr Brian Hughes
              School of Psychology

brian.hughes@nuigalway.ie            @b_m_hughes
Evidentiary reasoning:
Why do people believe weird
          things?
Difficulties with probabilistic reasoning

Tossing one coin, what is the chance
of 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)
Difficulties with probabilistic reasoning

Tossing two coins, what is the
chance 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)
Difficulties with probabilistic reasoning

Imagine you are at a party, where there are 23 people
present (including yourself). What are the chances that
two 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)
Difficulties with probabilistic reasoning

Imagine you are at a party, where there are 23 people
present (including yourself). What are the chances that
two 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)
Difficulties with probabilistic reasoning

If a test to detect a disease whose prevalence is 1/1000 has a false
positive rate of 5%, what is the chance that a person found to have a
positive result actually has the disease, assuming you know nothing
about 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)
Difficulties with probabilistic reasoning

If a test to detect a disease whose prevalence is 1/1000 has a false
positive rate of 5%, what is the chance that a person found to have a
positive result actually has the disease, assuming you know nothing
about 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)
Difficulties with probabilistic reasoning

If a test to detect a disease whose prevalence is 1/1000 has a false
positive rate of 5%, what is the chance that a person found to have a
positive result actually has the disease, assuming you know nothing
about 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)
Difficulties with probabilistic reasoning

If a test to detect a disease whose prevalence is 1/1000 has a false
positive rate of 5%, what is the chance that a person found to have a
positive result actually has the disease, assuming you know nothing
about 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)
Difficulties with probabilistic reasoning

If a test to detect a disease whose prevalence is 1/1000 has a false
positive rate of 5%, what is the chance that a person found to have a
positive result actually has the disease, assuming you know nothing
about the person’s symptoms or signs?


≈ 0.02     (or, in other words, a 1-in-50 chance)




                         Cited by Pinker (1997)
Difficulties with probabilistic reasoning

If a test to detect a disease whose prevalence is 1/1000 has a false
positive rate of 5%, what is the chance that a person found to have a
positive result actually has the disease, assuming you know nothing
about 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)
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
Difficulties with probabilistic reasoning
Difficulties with probabilistic reasoning
Difficulties with probabilistic reasoning
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
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
Probability of coincidences
What is the probability that you were
born in the same month as Barack
Obama?
     August
        1 in 12 (or 0.08)
What is the probability that you were
born on the same day-of-the-week as
Barack Obama?
     Friday
         1 in 7 (or 0.14)
What is the probability that you share            Born 1961
your 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
Probability of coincidences

What 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)
PS409
   Psychology, Science,
     & Pseudoscience



           Dr Brian Hughes
              School of Psychology

brian.hughes@nuigalway.ie            @b_m_hughes

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

  • 1. PS409 Psychology, Science, & Pseudoscience Dr Brian Hughes School of Psychology brian.hughes@nuigalway.ie @b_m_hughes
  • 2. Evidentiary reasoning: Why do people believe weird things?
  • 3. Difficulties with probabilistic reasoning Tossing one coin, what is the chance of 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. Difficulties with probabilistic reasoning Tossing two coins, what is the chance 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. Difficulties with probabilistic reasoning Imagine you are at a party, where there are 23 people present (including yourself). What are the chances that two 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. Difficulties with probabilistic reasoning Imagine you are at a party, where there are 23 people present (including yourself). What are the chances that two 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. Difficulties with probabilistic reasoning If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about 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. Difficulties with probabilistic reasoning If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about 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. Difficulties with probabilistic reasoning If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about 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. Difficulties with probabilistic reasoning If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about 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. Difficulties with probabilistic reasoning If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs? ≈ 0.02 (or, in other words, a 1-in-50 chance) Cited by Pinker (1997)
  • 12. Difficulties with probabilistic reasoning If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about 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. 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
  • 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. 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. Probability of coincidences What is the probability that you were born in the same month as Barack Obama? August 1 in 12 (or 0.08) What is the probability that you were born on the same day-of-the-week as Barack Obama? Friday 1 in 7 (or 0.14) What is the probability that you share Born 1961 your 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. Probability of coincidences What 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. PS409 Psychology, Science, & Pseudoscience Dr Brian Hughes School of Psychology brian.hughes@nuigalway.ie @b_m_hughes