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

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

14. Difficulties with probabilistic reasoning

15. Difficulties with probabilistic reasoning

16. Difficulties with probabilistic reasoning

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