The Sociology of Risk
Objective Probabilities
Goals:
1) To understand how to calculate objective probabilities
2) To recognize that our understanding of objective
probabilities is shaped by our perceptions
Risk and Uncertainty
• Arnoldi defines ‘risk’ as potential dangers
• Risk usually refers to negative events, but ‘risk’ is
sometimes used in a generic ways for positive and
negative events
• Probability is implied in the definition (i.e., ‘potential’)
• ‘Risk’ refers to a situation where the probability of the
event can be calculated
• ‘Uncertainty’ refers to a situation where the probability
of the event cannot be calculated
• There is a difference between objective and subjective
probabilities
Subjective probability
• Subjective probabilities depend on the person
making the assessment
• Compare a coin flip to a horse race – what is
the difference?
• Some events/games are not repeatable
Objective Probability
• “An objective probability is a probability that everyone
agrees on” (Amir D. Aczel. 2004. Chance: A Guide to
Gambling, Love, the Stock Market, & Just About
Everything Else.)
• Probability is the ratio of the number of times the
desired outcome can occur relative to the total number
of all outcomes that can occur over the long run.
• Probabilities are often expressed as ratios and/or
proportions.
Coin example
• The probability of a ‘heads’ on one flip of an
honest coin=1/2=0.5
• The flip of a coin is a purely random event –
we cannot predict the outcome of one flip
with certainty
• We can predict that the proportion of heads
over many flips is 0.5
• As the number of flips increases, the
proportion will center on the value of 0.5
Coin example
• Elementary outcomes: heads, tails
• This is an equal probability process
0.6
0.5
Probability
0.4
0.3
0.2
0.1
0
Heads
Tails
Dice example
• The probability of a ‘6’ on one roll of an
honest die=1/6=0.1667
• The roll of a die is a purely random event – we
can’t predict the outcome of any one roll
• We can, however, predict that the proportion
of 6s over many rolls is about 0.1667
• As the number of rolls increases, the
proportion will center on 0.1667
Dice example
• Elementary outcomes: 1, 2, 3, 4, 5, 6
• This is an equal probability process
0.18
0.16
0.14
Probability
0.12
0.1
0.08
0.06
0.04
0.02
0
1
2
3
4
5
6
Basic rules for more complex events…
• The ‘or’ rule (when to add)
– The probability of a 1 or a 6 on one roll of an
honest die=1/6+1/6=2/6=0.333
• The ‘and’ rule (when to multiply)
– The probability of a 6 and a 6 on the roll of two
honest dice=1/6*1/6=1/36=0.0278
Two dice
36 Elementary outcomes
1,1
2,1
3,1
4,1
5,1
6,1
1,2
2,2
3,2
4,2
5,2
6,2
1,3
2,3
3,3
4,3
5,3
6,3
1,4
2,4
3,4
4,4
5,4
6,4
1,5
2,5
3,5
4,5
5,5
6,5
1,6
2,6
3,6
4,6
5,6
6,6
Sum
Die 1
Die 2
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
11 Possible totals Expected frequency Expected proportion
2
1
0.0278
3
2
0.0556
4
3
0.0833
5
4
0.1111
6
5
0.1389
7
6
0.1667
8
5
0.1389
9
4
0.1111
10
3
0.0833
11
2
0.0556
12
1
0.0278
Sum
36
1
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Coin example
• Four flips of an ‘honest’ coin and the number
of heads
• This is a binomial process
• Both outcomes (heads and tails) are equally
likely so there is a uniform distribution for one
flip
• There is NOT a uniform distribution for
multiple flips – with 4 flips there are 5 possible
outcomes but 16 ways to get them
How many heads on four flips of an honest coing?
Probability
0.0625 0.2500 0.3750 0.2500 0.0625
# Heads
0
1
2
3
4
0.0625 0.0625 0.0625 0.0625 0.0625
Ways
1
4
6
4
1
TTTT
HTTT
HHTT HHHT HHHH
THTT
HTHT HHTH
TTHT
HTTH HTHH
TTTH
THHT THHH
THTH
TTHH
0.4
0.35
Probability
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
Number of Heads on 4 Flips of a Coin
Other Games – the Lottery
• It is possible to calculate the objective
probability of many games…
• Lottery (6 balls numbered 1-54)
• You need the 1 correct set of six numbers out
of the 25,827,165 possible unique
combinations of six numbers
• …So don’t play the lottery
Roulette
• A roulette wheel has 38 buckets:
–
–
–
–
36 numbers (1-36)
0 (green)
00 (green)
2 colors for the 36 numbers: red and black
• The house has a built-in advantage (because of how it sets
the bets – for example, 35 to 1 for a single number); it wins
over the long run
Bet
United States Roulette Rules
Pays
Probability
Win
1
47.37%
1
47.37%
1
47.37%
1
47.37%
1
47.37%
1
47.37%
2
31.58%
2
31.58%
2
31.58%
Red
Black
Odd
Even
1 to 18
19 to 36
1 to 12
13 to 24
25 to 36
Sixline (6
numbers)
First five (5
numbers)
Corner (4
numbers)
Street (3 numbers)
House
Edge
5.26%
5.26%
5.26%
5.26%
5.26%
5.26%
5.26%
5.26%
5.26%
5
15.79%
5.26%
6
13.16%
7.89%
8
10.53%
5.26%
11
7.89%
5.26%
17
5.26%
5.26%
35
2.63%
5.26%
Split (2 numbers)
Any one number
Blackjack
• What makes Blackjack so interesting is that it is based on
continuous probability; the probabilities actually change during play
with each passing card…it is a game with a memory; this is what
makes it possible to beat the casino (assuming an auto shuffler is
not used after every hand with replacement)
• Basic rules
–
–
–
–
–
–
–
Closest to 21 wins; tie=‘push’
Ace=1 or 11
2-9=face value
10, J, Q, K=10
Dealer must hit until sum totals 17, 18, 19, 20, or 21
Blackjack=21 on two cards; beats all but dealer blackjack, pays 3 to 2
Plays: hit, stay, split, double down (double bet for 1 card), surrender
(you get 50% of your bet back), insurance (costs half of current wager,
pays 2 to 1 if Blackjack 21)
Blackjack
•
The longer you play, the greater the likelihood that you will lose everything, but
you can improve your chances by following basic strategy…
Blackjack
• Counting cards – the Hi-low System
– Developed by MIT Professor Edward Thorp
• Simulations show that when low cards (7 and under) are left
in the deck, the odds favor the dealer; high cards (9 and up)
favor the player
• The Hi-lo system is based on a running tally, not memorizing
every card
–
–
–
–
–
+1: 2-6
-1: 10, J, Q, K, Aces
7, 8, 9 are not counted
You increase your bet when the count is high
An equation determines how much to raise the bet
(count/number of decks not seen); also incorporate house
advantage
Probabilities are confusing
• Calculating objective probabilities is not always easy, but it can be done
• Despite this, many problems/games confound people because they are
non-intuitive
• Monty Hall Problem / Let’s Make a Deal
–
–
–
–
–
http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
Contestant selects one of three doors
From the remaining two doors, the host selects one non-winning door
The contestant is asked: stay with their original selection or change doors?
What should they do?
• Probability of picking the winning door is 1/3
• Probability of not picking the winning door is 2/3
• When one of the non-selected doors is revealed, the probability for the two non-selected
is still 2/3
• So the probability of switching and winning is 2/3
• The probability of not switching and winning is still 1/3
Monte Hall
Door 1
Door 2
Door 3
result if switching result if staying
Car
Goat
Goat
Goat
Car
If stay, win 1/3
Goat
Car
Goat
Car
Goat
If switch, win 2/3
Goat
Goat
Car
Car
Goat
Initial pick is always Door 1
vos Savant, Marilyn (1990). "Ask Marilyn" column, Parade Magazine p. 16
(9 September 1990).
Paul the Octopus
• Paul the octopus picked the winner of 8
straight soccer matches in 2010 FIFA World
Cup
• Is Paul some sort of soccer genius?
Paul the Octopus
0.5
# games correct in a row
Probability
How many animals would need to make
predictions for 1 to be correct 8/8?
1
0.50000000000
2
2
0.25000000000
4
3
0.12500000000
8
4
0.06250000000
16
5
0.03125000000
32
6
0.01562500000
64
7
0.00781250000
128
8
0.00390625000
256
You didn’t hear about the other 255 animals that didn’t predict as well as Paul…
Perceptions of Objective Probabilities
• Things casinos do:
– Pictures of recent big winners with checks
– Slots that flash, have sirens, and falling coins – these draw our
attention
– Use of chips instead of US Dollars
• Biases:
– We have selective memory
– We suffer from hopeful thinking
– We use faulty logic
• Gambling versus counting
– Good ‘cheats’ don’t always win; they utilize information to
increase their chances
– They must be subtle or it won’t work