1. Lazy and Fallacious Thinking in Statistics “Let’s Make a Deal” Chris Mellor

2. Objective
To introduce & discuss - largely by example - some situations which are commonly encountered and highlight the problems (and incorrect inferences that may result) from ‘lazy thinking’ and a partial understanding of probability

3. Contents Probability and conditional probability
The Monty Hall problem
Lazy Thinking: System 1 and System 2
The Conjunction Fallacy
Randomness & issues of Mix
The Base Rate Fallacy
Anchors.

4. Probability
There are situations which appear on the surface to be quite straightforward extensions of basic probability ideas which have caused problems for many (not just undergraduates).
One of the most famous examples is probably the ‘Monty Hall’ problem, from a late 1950’s US game show called ‘Lets Make a Deal’ – but before we discuss this let’s look at a ‘simpler’ example by way of introduction……

5. Example - 1
Consider a game where there are 3 coins in a bag. The coins are respectively W, B, B, identical in size and feel.
After agreeing a (non-returnable) stake, you can pick a coin from the bag. If you pick the white coin you win £10.
What would you be prepared to stake to play the game?

6. Example - 2
If you were a ‘rational’ agent you would probably reason like this….
Each coin is equally likely to be picked (probability 1/3) so presuming you get £10 for picking the W coin (true), the “expected payoff” from playing the game is
(10* 1/3) + (0*1/3) + (0*1/3) = £3.33
Effectively a weighted average of the payoff’s (each weight being the 1/3 equally likely chance of picking the coin)

7. Example - 3
So if you were asked to pay (say) £2 to play, you might suspect (correctly) there is money to be made ….. and ……..
If you played the game long enough each ‘game’ would cost you £2 and on average and in the long term you would get a return of £3.33 / game, so over a period of time you might expect to make money (and retire before graduating).

8. The Monty Hall Game show - outline
After several rounds when contestants are ‘knocked out’ the game show eventually produces a ‘winner’.
In the final part of the show, the ‘winning’ contestant is offered the choice of 3 ‘identical’ doors. Behind 2 of the doors is a goat, while the star prize (a car) is behind the third door.
The contestant chooses a door.
The game show host (who knows the
location of the prizes) then opens one
of the 2 remaining doors to reveal a goat
(this is always possible).
The contestant is then offered the chance to
switch his door for the remaining unopened
door. What would you do?

9. Monty Hall from the Movie 21

10. The Monty Hall Paradox
The optimal strategy is for the contestant to ALWAYS switch to the remaining door because if they stick
P(win) = 1/3 (they chose this door from 3 equally likely to contain the car), and this is still the case.
We know one of the other doors has been opened (and revealed a goat) so if the probability that our selected door hides the car is 1/3 – the probability that the other unopened door MUST hide the car is 1-1/3 = 2/3.
(After all the car is behind one of the 2 remaining doors)

11. The Monty Hall Paradox, continued
The flawed thinking goes something like this...
When I first picked a door the chance is contained the car was 1/3 (correct).
The fact that I’ve been shown a door with a goat, means there are now 2 doors left, and each is equally likely to have the car behind it (incorrect)
So, and following on from this, my chance of having the car has gone up from 1/3 to ½ (the extra information of the door with the goat has increased my chances). (incorrect)
So it doesn’t matter if I switch or not, and I’d kick myself if I had originally picked the car and swapped it for the goat, so I’ll stick with my original choice.

12. The Monty Hall Paradox, continued
If there still lingers any doubt, consider the situation where there are not 3 doors but 100 doors.
Similar rules. But after picking, the game show host opens all but 1 of the remaining 99 doors (98) – all showing goats. You are now faced with a choice. The probability the selected door wins the car is 1/100. The probability that the car is behind the ‘other’ unopened door is 99/100. So MUCH better to switch.

13. Conclusions
The Monty Hall paradox makes you think of alternative ways to view a problem and not to blindly jump to the ‘most obvious / gut feel’ answer .
Logical thought can lead to ‘the correct’ strategy, and where the ‘additional’ information (that the goat is NOT behind one of the doors) will ALWAYS lead you to an improved situation which you can take advantage of by switching your initial choice.

14. Back to our 3 coin example..
You have agreed that a fair stake to play the game as set up is £3.33 and that a rational agent would play if the stake were less than this (say £3 / game).
You set this up so that a colleague picks a coin – as before, but without looking at it – but this time you offer to show them one of the coins they did not select. [Moreover you ALWAYS show them a B coin – one of the losing coins.] But the cost to them of this ‘extra’ information is that the stake rises to £4.

15. Example - continued
Some will incorrectly reason that the additional information is ‘worth’ buying because it reduces the number of coins to 2 and accordingly the chance of them winning increases to 0.5.
There are now only 2 possible outcomes – you win or you lose depending on the coin you hold.
They may also reason that if this is true then it is worth paying up to £5 to play and a £4 stake is a good deal.
If they take the bait you have a chance to make money.

16. Example - continued
If they refuse to play, then offer them the ‘opportunity’ to make it more interesting by actually showing them a (B) coin, without any stake increase, but ….
Now offer to switch coins but tell them since there is clearly one B and one W coin left, you will
pay out (£10) now if he holds the Black coin
after the switch. And the stake is now £4 up
from £3.33 but lower than the £5 which
you hope he thinks makes it a fair game.
If he bites you win…(most of the time)…

17. Lazy Thinking - 1
The underlying problem is that we are all naturally lazy and that (unless we are primed) we default to our System 1 way of thinking
Because it’s easier
Because it requires less effort
Because we’re busy
(sometimes) because we’re
pre-occupied with doing other things

18. Lazy Thinking - 2
But System 1 CAN dominate if you allow it to (it is for many of us , the default)– and you sometimes have to make a conscious effort to engage your System 2 to address a situation.
Take the following example and I’m going to ask one of you in a moment to give me your answer….you will have around 5 seconds……

19. Lazy Thinking - 3
Work out (in your head)
23 * 17

20. Lazy Thinking - 4
As humans we need both Systems to function and we need a balance to ensure we stay alive, BUT in a busy world System 1 (the lazy one) has a tendency to (if not dominate) push out System 2 ESPECIALLY when
We’re tired
We’re preoccupied or busy
We’re juggling several things at once
We’re timed constrained
We’re ‘tricked’ (via, e.g. Anchors)

21. Example (The Müller-Lyer illusion)
Which of the 2 horizontal lines is longest?

22. Underlying Data

23. Treasury Chart (7/2/14) - as reported by BBC

25. The Conjunction Fallacy
This example was one that Kahneman used (and has been widely used since to illustrate what is known as the Conjunction Fallacy – simply often referred to as Linda).
Linda is 31 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 anti-nuclear demonstrations.
Which is more probable?
• Linda is a bank teller.
• Linda is a bank teller and is active in the feminist movement.

27. In one experiment the question of the Linda problem was reformulated as follows:
There are 100 persons who fit Linda’s description. Estimate how many of them are:
Bank tellers? __ of 100
Bank tellers and active in the feminist movement? __ of 100
Whereas previously 85% of participants gave the wrong answer (bank teller and active in the feminist movement), in experiments done with this questioning none of the participants gave a ‘wrong’ answer.

28. Exercise
A very large urn with millions of balls in it, contains only RED or BLUE balls. 50% of the balls are RED and 50% are BLUE.
A sequence of balls are drawn from the urn. Which of the following sequences of 9 balls is most likely, which least likely.
R R R R R R R R R
R B R R B B R B R

29. Which hospital to close?
A regional hospital trust is considering closing the Cardiac Department of one of 2 hospitals in the region. The decision of which to close depends on ‘survival’ rates from bypass operations carried out at each hospital in the previous year. The hospital with the higher (survival) rate (lower mortality rate) keeps its Cardiac Unit – the other is to close.
Hospital A undertook 2100 operations, with 63 deaths (3%).
Hospital B undertook 2400 operations, with 48 deaths (2%).

30. Obviously….. Hospital A should close.
But Hospital A put in a last minute appeal…
They argued that if the data were analysed by gender the reverse decision would be made.

31. Summary Hospital A Hospital B TOTAL F M Total No of Operations 2100
600
1500
2400
1800
Deaths 63 6 57 48 24
Rate
3.0%
1.0%
3.8%
2.0%
1.3%
4.0%

32. Base Rates - 1 In many situations we face we ignore ‘base rates’.
We either don’t search for information …..or … even armed with the additional information we don’t know how to incorporate it with the other evidence to make a better decision.
e.g. Suppose a police force have a breathalyser which reads positive in 5% of cases where the driver is in fact under the limit. The test is 100% reliable in that it NEVER fails to detect a drunk driver.
Estimate the probability that a driver stopped at random who fails the test is really over the limit?

33. Base Rate Fallacy - 2
Most people faced with this ‘ignore’ the fact that there are in reality very few drunk drivers (assume it’s 1 in 1000)
Someone who is stopped therefore is VERY unlikely to be drunk AND they are very unlikely to fail the test if they are sober (1/20).
We should use BAYES THEOREM to calculate the probability because we are combining prior information and evidence…

34. Base Rate Fallacy - 3 Here Prob(drunk / + breathaliser)
= Prob(+ breathaliser / drunk). Prob(drunk)
Prob( + breathaliser)
= * =

35. Anchors / Framing - 1
On many occasions we get heavily influenced by (sometimes erroneous) information – often based on hearsay or dubious sources (often not traceable or reproducible). These become part of everyday life – especially in organizations – and acquire an almost folklore status.
We drop anchors all the time – sometimes inadvertently, sometimes deliberately (e.g. in negotiations)

36. Anchors / Framing - 2
Astute lawyers wishing to cast doubt on a witnesses character may (knowingly) make a statement hidden in a question which they KNOW will be struck from the record. The judge may instruct the jury to ignore the question, but the damage is done....
[ + Brexit claims….]
e.g. Alex recently retired. How old do you estimate Alex to be?

37. Example Was Gandhi older than 35 when he was assassinated?
How old was Gandhi when he was killed?

38. Summary
The examples and discussion should lead you to be very wary when making inferences or using what you might consider to be intuitive reasoning to make decisions.
The apparently rational decision may not be the ‘correct’ one (eg Monty Hall Problem)
Your System 1 might be over-riding your System 2 (you are tired or you ignore vital information)
You may be tricked by Illusions
You might be being influenced by anchors.
BEWARE