The Sociology of Risk

Objective ProbabilitiesGoals: 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

Heads Tails0

0.1

0.2

0.3

0.4

0.5

0.6Pr

obab

ility

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

1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Prob

abili

ty

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 dice36 Elementary outcomes1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6

SumDie 1

1 2 3 4 5 6Die 2 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 5 6 7 8 9 10 11 6 7 8 9 10 11 12

11 Possible totals Expected frequency Expected proportion2 1 0.02783 2 0.05564 3 0.08335 4 0.11116 5 0.13897 6 0.16678 5 0.13899 4 0.1111

10 3 0.083311 2 0.055612 1 0.0278

Sum 36 1

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 0.0625Ways 1 4 6 4 1

TTTT HTTT HHTT HHHT HHHHTHTT HTHT HHTHTTHT HTTH HTHHTTTH THHT THHH

THTHTTHH

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Number of Heads on 4 Flips of a Coin

Pro

ba

bili

ty

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

United States Roulette RulesBet Pays Probability

WinHouse Edge

Red 1 47.37% 5.26%Black 1 47.37% 5.26%Odd 1 47.37% 5.26%Even 1 47.37% 5.26%

1 to 18 1 47.37% 5.26%19 to 36 1 47.37% 5.26%1 to 12 2 31.58% 5.26%

13 to 24 2 31.58% 5.26%25 to 36 2 31.58% 5.26%

Sixline (6 numbers) 5 15.79% 5.26%

First five (5 numbers) 6 13.16% 7.89%Corner (4 numbers) 8 10.53% 5.26%Street (3 numbers) 11 7.89% 5.26%

Split (2 numbers)17 5.26% 5.26%

Any one number35 2.63% 5.26%

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 HallDoor 1 Door 2 Door 3 result if switching result if staying

Initial pick is always Door 1

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

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# games correct in a row Probability

How many animals would need to make predictions for 1 to be correct 8/8?

0.5 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