### Critical Thinking

```Critical Thinking
All of the following examples came
from Daniel Kahneman’s book Thinking,
Fast and Slow.
Question #1
Here is a simple puzzle.
Do not try to solve it but listen to your intuition:
A bat and ball cost \$1.10.
The bat costs one dollar more than the ball.
How much does the ball cost?
The ball costs \$0.05.
Question #2
Try to determine, as quickly as you can, if the
argument is logically valid. Does the conclusion
All roses are flowers.
It is possible that there are no roses among the
Question #3
If it takes 5 machines 5 minutes to make 5
widgets,
how long would it take 100 machines to make
100 widgets?
100 minutes or 5 minutes?
5 minutes
Question #4
In a lake, there is a patch of lily pads. Every day,
the patch doubles in size. If it takes 48 days for
the patch to cover the entire lake, how long
would it take for the patch to cover half of the
lake?
24 days or 47 days?
47 days
#5
Question 6 for group A
How happy are you these days?
How many dates did you have last month?
On a scale of 1 to 10 with 10 being the happiest,
how happy are you?
Question 6 for group B
How many dates did you have last month?
How happy are you these days?
On a scale of 1 to 10 with 10 being the happiest,
how happy are you?
Question #7
Take the sex of six babies born in sequence at a
hospital. Consider three possible sequences:
BBBGGG
GGGGGG
BGBBGB
Which sequence is most likely?
Because the events are independent and
because the outcomes B and G are
(approximately) equally likely, then any possible
sequence of six births is as likely as any other.
Question 8 for group A
Anchoring
Is the height of the tallest redwood more or less
than 1,200 feet?
tallest redwood?
Question 8 for group B
Anchoring
Is the height of the tallest redwood more or less
than 180 feet?
tallest redwood?
Question 9
Consider the pairs of causes of death. Indicate
the more frequent cause and estimate the ratio
of the two frequencies.
Strokes and Accidents
Lightning and Botulism
Disease and Accidents
Diabetes and Accidents
Strokes cause almost twice as many deaths as all accidents
combined.
Asthma causes 20 times more deaths than tornadoes
Death by lightning is 52 more frequent than by botulism
Death by disease is 18 times more likely as accidental life
Death by diabetes is 4 times more likely than by accident
Question #10
Computer science
Engineering
Humanities and education
Pre-Law
Library science
Physical and life sciences
Social science & social work
The following is a personality sketch of Tom W
written during Tom’s senior year in high school
by a psychologist, on the basis of psychological
tests of uncertain validity:
Tom W is of high intelligence, although lacking in
true creativity. He has a need for order and
clarity, and for neat and tidy systems in which
every detail finds its appropriate place. His
writing is rather dull and mechanical,
occasionally enlivened by somewhat corny puns
and flashes of imagination of the sci-fi type. He
has a strong drive for competence. He seems to
have little feel and little sympathy for other
people, and does not enjoy interacting with
others. Self-centered, he nonetheless has a deep
moral sense.
Now please take a piece of paper and rank the
fields of specialization in order of the likelihood
that Tom W is now a student in each of these
fields.
Tom W was intentionally designed as an “antibase- rate” character, a good fit to small groups
and a poor fit to the most populated specialties.
Substitution is perfect in this case: there is no
indication that participants did anything else but
probability (likelihood) was difficult, but the
question about similarity was easier, and people
Computer science
Engineering
Humanities and education
Pre-Law
Library science
Physical and life sciences
Social science & social work
17.3 % of students
2.8
5.3
24.8
4.9
0.9
13.2
8.6
Question #11
Linda is thirty-one years old, single, outspoken, and very
bright. She majored in philosophy. As a student, she was
deeply concerned with issues of discrimination and social
justice, and also participated in antinuclear
demonstrations.
Which statement is more likely correct
Linda is a bank teller who is active in the feminist
movement
Linda is a bank teller
Think in terms of Venn diagrams.
The set of feminist bank tellers is wholly
included in the set of bank tellers, as every
feminist bank teller is a bank teller.
Question #12
Price sets of dinnerware offered in a local store, where
dinnerware regularly runs between \$30 and \$60.
Assuming the dishes in the two sets are of equal quality.
Please set a price for Set A and a price for Set B.
Please set a price for Set A and a price for Set B.
Question #13
A regular six-sided die has four green faces and to
red faces will be rolled 20 times. Which sequence
will be more likely?
RGRRR
GRGRRR
GRRRRR
Because the die has twice as many green sides as red
sides, the first sequence is quite unrepresentative.
The second sequence, which contains six tosses, is a
better fit to what we expect from this die, because it
includes two G’s.
However, this sequence was constructed by adding a G to
the beginning of the first sequence, so it can only be less
likely than the first.
Example 14
A cab was involved in a hit and run accident at night.
Two companies, the Green and Blue, operate in the
city. You are given the following data:
Example 14
85% of the cabs in the city are Green and 15% Blue.
A witness identified the cab as Blue. The court tested
the reliability of the witness under the circumstances
that existed on the night of the accident and concluded
that the witness correctly identified each of the two
colors 80% of the time and failed 20% of the time.
What is the probability the cab involved in the accident
was Blue?
′
Bayes Theorem  |′ ′ =
=
′| ×
′
′| ×
′ | ×  +  ′ | ×
.80×.15
.80×.15+ .20 ×.85
OR
12
12 +17
baseline
Blue
Green
B
G
witness Blue B’
12
17
Green G’
3
68
15
85
reports
29
Overconfidence #15
For a number of years, professors at Duke
conducted a survey in which CFO’s of large
corporations estimated the returns of the S&P
index over the following year. The Duke scholars
collected 11,600 such forecasts and examined
their accuracy.
Overconfidence #15
The conclusion was straightforward: CFOs of
larger corporations had no clue about the shortterm future of the stock market; the correlation
between their estimates and the true value was
slightly less than zero!
The truly bad news is that the CFOs did not
appear to know that their forecasts were
worthless.
Overconfidence #15
In addition to their estimates, participants
provided two other estimates: a value that they
were 90% sure would be too high, and one that
they were 90% sure would be too low. The range
between the two values is called an “80%
confidence interval” and outcomes that fall
outside the interval are labeled surprises. …
Overconfidence #15
… As frequently happens in such exercises, there
were far too many surprises; their incidence was
67%, more than 3 times higher than expected.
This shows that CFOs were grossly overconfident
about their ability to forecast the market.
Overconfidence #15
The confidence interval that properly reflects
the CFOs knowledge is more than four times
wider than the intervals they actually stated.
Gamble #16
Anthony’s current wealth is 1 million
Betty’s current wealth is 4 million.
They are both offered a choice between a
gamble and a sure thing.
The gamble:
equal chances to end up owning 1 or 4 million
OR The sure thing: own 2 million for sure
Gamble #16
Anthony and Betty face the same outcomes:
their expected wealth will be b 2.5 million if
they take the gamble and 2 million if they prefer
the sure-thing option.
Gamble #16
Anthony (who currently owns 1 million):
If Anthony chooses the sure thing, his wealth
will double with certainty. This is very attractive.
Alternatively, he can take a gamble with equal
chances to quadruple his wealth or gain nothing.
Gamble #16
Betty (who currently owns 4 million):
If Betty chooses the sure thing, she loses half of
her wealth with certainty, which is awful.
Alternatively, she can take a gamble with equal
chances to lose three quarters of her wealth or
to lose nothing.
Gamble #17 part 1
In addition to whatever you own, you have been
given \$1,000.
You are now asked to choose one of these
options:
50% chance to win \$1,000
OR get \$500 for sure
Gamble #17 part 2
In addition to whatever you own, you have been
given \$2,000.
You are now asked to choose one of these
options:
50% chance to lose \$1,000
OR lose \$500 for sure
Gamble #17
The outcomes are identical
50% \$2,000
50% \$1,000
Or
\$1,500 with certainty
A: 61% chance to win \$520,000 OR 63% chance to win \$500,000
B: 98% chance to win \$520,000 OR 100% chance to win \$500,000
Imagine the outcome will be determined by a blind draw
from an urn containing 100 marbles – you win if you draw
a red marble, you lose if you draw white.
A: 61% chance to win \$520,000 OR 63% chance to win \$500,000
B: 98% chance to win \$520,000 OR 100% chance to win \$500,000
In problem A, almost everyone prefers the left-hand urn,
although it has fewer winning red marbles, because the
difference in the size of the prize is more impressive than
the difference in the chances of winning.
In problem B, a large majority choose the urn with that
guarantees a gain of \$500,000.
A: 61% chance to win \$520,000 OR 63% chance to win \$500,000
B: 98% chance to win \$520,000 OR 100% chance to win \$500,000
The two urns in problem B are more favorable versions of
problem A, with 37 white marbles replaced by red
winning marbles in each earn. The improvement of the
left is clearly superior because each red marble gives you
a chance to win \$520,000 on the left and only \$500,000
on the right. So you started in the first problem with a
preference for the left-hand urn, which was improved
more than the right-hand urn – but now you prefer the
urn on the right.
A: 61% chance to win \$520,000 OR 63% chance to win \$500,000
B: 98% chance to win \$520,000 OR 100% chance to win \$500,000
A: 61% chance to win \$520,000 OR 63% chance to win \$500,000
C: 37% chance to win \$520,000 OR 37% chance to win \$500,000
= 98% chance to win \$520,000 OR
100% chance to win \$500,000
NBA playoffs #19
When I wrote this the following teams had the 8
best records in the NBA
Miami Heat
San Antonio Spurs
Oklahoma City Thunder
LA Clippers
Memphis Grizzlies
Denver Nuggets
Indiana Pacers
Brooklyn Nets
NBA playoffs
Estimate the probability the following team will
win the NBA playoffs?
Miami Heat
San Antonio Spurs
Oklahoma City Thunder
LA Clippers
Memphis Grizzlies
Denver Nuggets
Indiana Pacers
Brooklyn Nets
Concurrent decisions #20
Imagine that you face the following pair of
concurrent decision.
First examine both decision, then make your
choices.
Concurrent decisions #20
Decision (i): Choose between
Sure gain of \$240
25% chance - gain \$1,000 and 75% chance - gain nothing
Decision (ii): Choose between
Sure loss of \$750
75% chance - lose \$1,000 and 25% chance - lose nothing
Concurrent decisions #20 combined
AC: sure loss \$510
AD: 25% win \$240 and 75% lose \$760
BC: 25% win \$250 and 75% lose \$750
BD 6.25% win \$1,000, 37.5% \$0, 56.25% lose \$1,000
EV(AC) = loss \$510
EV(BC) = loss \$500
EV(BD) = loss \$500
A repeated gamble #21
The great Paul Samuelson famously asked a friend
whether he would accept a gamble on the toss of
a coin in which he could lose \$100 or win \$200.
Would you take such a gamble?
Would you take a similar gamble if you were
allowed to make 100 such bets?
One Bet 50% chance you win \$200
50% chance you lose \$100
Two Bets 25% chance you win \$400
50% chance you win \$100
25% chance you lose \$200
Three Bets 12.5% chance you win \$600
37.5% chance you win \$300
37.5% chance you break even
12.5% chance you lose \$300
A repeated gamble #21
According to the book there is 1/2,300 chance of
losing any money.
Notice 34 heads and 66 tails results in a \$200 gain
And 33 heads and 67 tails results in a \$100 loss
Victim Compensation 22 (group A)
You have the task of setting compensation for
victims of violent crimes. You consider the case
of a man who lost the use of his right arm as a
result of a gunshot wound. He was shot when
he walked in on a robbery occurring in a
convenience store in his neighborhood.
Victim Compensation 22 (group A)
Two stores were located near the victim’s
home, one of which he frequented more
regularly than the other.
The burglary happened in the man’s regular
store.
Victim Compensation 22 (group B)
Two stores were located near the victim’s
home, one of which he frequented more
regularly than the other.
The man’s regular store was closed for a funeral,
so he did his shopping in the other store.
Roulette Wheel 23
You are offered a choice between two bets,
which are to be played on a roulette wheel with
36 sectors.
Bet A: 11/36 to win \$180, 25/36 to lose \$15
Bet B: 35/36 to win \$40, 1/36 to lose \$10
Roulette Wheel 23
Bet A: 11/36 to win \$180, 25/36 to lose \$15
Bet B: 35/36 to win \$40, 1/36 to lose \$10
EV(A) = \$44.58
EV(B) = \$38.61
Epidemic 24 (group A)
Imagine that the United States is preparing for
the outbreak of an unusual Asian disease, which
is expected to kill 600 people. Two alternative
programs to combat the disease have been
proposed. Assume that the exact scientific
estimates of the consequences of the programs
are as follows:
Epidemic 24 (group 1)
If program A is adopted, 200 people will be
saved.
If program B is adopted, there is a one-third
probability that 600 people will be saved and a
two-thirds probability that no people will be
saved.
Epidemic 24 (group 2)
If program A is adopted, 400 people will die.
If program B is adopted, there is a one-third
probability that nobody will die and a two-thirds
probability that 600 people will die.
Epidemic 24
The outcomes are exactly the same for Group 1
and Group2. They are simply framed differently.
Theater Tickets #25 (group A)
A woman has bought two \$80 tickets to the
theater. When she arrives at the theater, she
opens her wallet and discovers that the tickets
are missing. Will she but two more tickets to see
the play?
Theater Tickets #25 (group B)
A woman goes to the theater, intending to buy
two tickets that cost \$80 each. She arrives at the
theater, opens her wallet, and discovers to her
dismay that the \$160 with which she was going
to make the purchase is missing. She could use
her credit card. Will she buy the tickets?
Gas Mileage #26
Consider two car owners who seek to reduce their
costs:
Adam switches from a gas-guzzler of 12 mpg to a
slightly less voracious guzzler that runs at 14 mpg.
The environmentally friendly Beth switches from a
30 mpg car to one that runs at 40 mpg.
Suppose both drivers travel equal distances over a
year. Who will save more gas by switching?
Gas Mileage #26
If both car owners drive 10,000 miles,
Adam will reduce his consumption from 833
gallons to 714 gallons, for a saving of 119
gallons.
Beth’s use of fuel will drop from 333 gallons to
250, saving only 83 gallons.
Gas Mileage #26
Decisions would improve if we used gallons per
mile rather than mpg.
Adam would reduce his usage by .0119 gpm (
.0833 gpm to .0714 gpm).
Beth would reduce her usage by .0083 gpm
(.0333 to .0250).
This number can easily be multiplied by the
number of miles driven.
Experience #27
Each participant experienced two cold-hand
episodes:
Experience #27
The short episode consisted of 60 seconds of
immersion in water at 14 degrees Celsius, which
is experienced as painfully cold, but not
intolerable. At the end of 60 seconds, the
experimenter instructed the participant to
remove his hand from the water and offered a
warm towel.
Experience #27
The long episode lasted 90 seconds. Its first 60
seconds were identical to the short episode. The
experimenter said nothing at all at the end of
the 60 seconds. Instead he opened a valve that
allowed slightly warmer water to flow into the
tub. During the additional 30 seconds, the
temperature of the water rose by roughly 1
degree, just enough for most subjects to detect
a slight decrease in the intensity of the pain.
Experience #27
Participants were told that they would have 3
cold-hand trials, but in fact they experienced
only 2 episodes. Trials were separated by 7
minutes. 7 minutes after the 2nd trial, the
participants were given a choice about the 3rd
trial. They were told that one of the experiences
would be repeated exactly, and they were free
to choose whether to repeat the short episode
or the long episode.
Experience #27
Fully 80% of the participants reported that their
pain diminished during the final phase of the
longer episode opted to repeat it. Thereby
declaring themselves willing to suffer 30
seconds of needless pain in the anticipated third
trial.
Final Thoughts
Nothing in life is as important as you think it is
when you are thinking about it.
Final Thoughts
How much pleasure do you get from your car”
When do you get pleasure from your car?
You get pleasure or displeasure from your car