- How to Be A Winner The Maths of Race Fixing
and
Money Laundering
John D Barrow
Why is Probability Theory Not Ancient?
Religious beliefs
Or
No concept of equally
likely outcomes
???
“And they said every one to his fellow, Come, and let us cast lots,
that we may know for whose cause this evil is upon us.
So they cast lots and the lot fell upon Jonah.”
Book of Jonah 1 v 7
St. Augustine: “We say that those causes that are said to be by chance are not
nonexistent but are hidden, and we attribute them to the will of the true God”
Astragali
Sheep's’ ankle bones, 6-sided, numbered, asymmetrical
Divination with sets of 5 in Asia Minor from 3600 BC
Eventually replaced by dice
Ancient Dice
The most popular dice game of the
Middle Ages: was called “hazard”
Arabic “al zhar” means “a die.”
Roman icosahedral die
20 faces
Right and Left-handed Dice
Western dice are right-handed: if the 1-spot is face up and the 2-spot
is turned to face the left then the 3-spot is to the right of it.
Chinese dice are left-handed: they will have the faces the opposite way round.
The Problem of the Points
Chevalier de Méré and Blaise Pascal and Pierre de Fermat
1654
Two people play a fair game
The first to win six points takes all the money.
How should the stakes be divided if the game
is interrupted when one has 5 points and the other 3?
HHH, HHT, HTH, TTT,
THT, TTH, THH. HTT
Player with 3 points has to win all the next 3 games.
He has 1/8 chance of doing that.
His opponent has a 7/8 chance of winning 1 more game.
Give 7/8 of prize money to the one with 5 and 1/8 to the other
More Chevalier de Méré
He won lots of money betting on at least 1 six in 4 rolls of a die
based purely on experience
Probability of no 6 is 5/6
Probability of no 6 in four throws is 5/6  5/6  5/6  5/6 = (5/6)4 = 625/1296
Probability of one 6 is 1 – 625/1296 = 671/1296 = 0.5177 > 1/2
So he thought that he should bet on one or more double 6’s
occurring in 24 rolls of 2 dice
Probability of no double sixes in 24 throws is (35/36)24 = 0.5086
Probability of one double six is 1 - (35/36)24 = 0.4914 < 1/2
After a while he stopped doing this !
Winning The Toss
Australian Open January 2008
Playing Fair With a Biased Coin
Unequal probability of H and T: p  ½
Probability of H is p
Probability of T is 1-p
Toss twice and ignore pairs HH and TT
Probability of HT is p(1-p)
Probability of TH is (1-p)p
Call combination HT ‘Newheads’
Call combination TH ‘Newtails’
Newheads and Newtails are equally likely
Efficiency is poor (50%) – discard the HH and TT s
Faking Random Sequences
1.
2.
3.
THHTHTHTHTHTHTHTHTTTHTHTHTHTHTHH
THHTHTHTHHTHTHHHTTHHTHTTHHHTHTTT
HTHHTHTTTHTHTHTHHTHTTTHHTHTHTHTT
Do these look like real random sequences ?
Some More Candidates
With 32 tosses
4. THHHHTTTTHTTHHHHTTHTHHTTHTTHTHHH
5. HTTTTHHHTHTTHHHHTTTHTTTTHHTTTTTH
6. TTHTTHHTHTTTTTHTTHHTTHTTTTTTTTHH
Are they random?
Some More Candidates
With 32 tosses
4. THHHHTTTTHTTHHHHTTHTHHTTHTTHTHHH
5. HTTTTHHHTHTTHHHHTTTHTTTTHHTTTTTH
6. TTHTTHHTHTTTTTHTTHHTTHHHHHHTTTTH
The chance of a run of r heads or r tails coming up is just
½  ½ ½  ½ …. ½, r times. This is 1/2r
If we toss our coin N > r times there are N different possible
starting points for a run of heads or tails
Our chance of a run of length r is increased to about N  1/2r
A run of length r is going to become likely when N  1/2r is roughly
equal to 1, that is when N = 2r.
Note that 32 = 25
Winning (and Losing) Streaks
The Nasser Hussain Effect
England cricket captain
“Flipping useless, Nasser!”
During 2000-2001
BBC
Atherton took over for one game after he had lost 7
and won the toss
Normal service was then resumed
There is a 1 in 214 = 16384 chance of losing all 14 tosses
But he captained England 101 times and there is a chance of about
1 in 180 of a losing streak of 14
Can You Always Win?
Or avoid ever losing ?
The Win-Win Scenario
The odds for the runners are a1 to 1, a2 to 1, a3 to 1, and so on, for any number of
runners in the race.
If the odds are 5 to 4 then we express that as an ai of 5/4 to 1
Bet a fraction 1/(ai +1) of the total stake money on the runner with odds of ai to 1
If there are N runners, we will always make a profit if
Q = 1/(a1 +1) + 1/(a2 +1) + 1/(a3 +1) +….+ 1/(aN +1) < 1
Winnings = (1/Q – 1)  our total stake
Example:
Four runners and the odds for each are 6 to 1, 7 to 2, 2 to 1, and 8 to 1 and.
Then we have a1 = 6, a2 = 7/2, a3 = 2 and a4 = 8 and
Q = 1/7 + 2/9 + 1/3 + 1/9 = 51/63 < 1
Allocate our stake money with 1/7 on runner 1, 2/9 on runner 2, 1/3 on runner 3,
and 1/9 on runner 4
We will win at least 12/51 of the money we staked (and of course we get our
stake money back as well).
Race Fixing ‘101’
The favourite is always the largest contributor to Q because a1 is
the smallest of the ai s
We could have Q > 1 with all runners included
Q = 1/(a1 +1) + 1/(a2 +1) +….. > 1
But if you know the favourite has been hobbled then you
calculate Q excluding a1 which can result in
Qfix = 1/(a2 +1) + 1/(a3 +1) + …. < 1
If there are 4 runners with odds
3 to 1, 7 to 1, 3 to 2, and 1 to 1
Q = 1/4 + 1/8 + 2/5 + 1/2 = 51/40 > 1
So we can’t guarantee a winning return
Dope the favourite and place you money on the other
three runners only, betting 1/4 of our stake money
on runner 1, 1/8 on runner 2, and 2/5 on runner 3
You are really betting on a 3-horse race with
Qfix = 1/4 + 1/8 + 2/5 = 31/40 < 1
Whatever the outcome you will never do worse
than winning your stake money plus
{(40/31) -1}  Stake money = 9/31  Stake money
When Bookies Disagree
Outcome
Bookmaker 1’s
odds
Bookmaker 2’s
odds
Oxford win
1.25
1.43
Cambridge
win
3.9
2.85
Q of Bookie 1 1.056 >1
He gains 5.6%
Q of Bookie 2 He gains 5.1%
1.051 > 1
A Mixed Strategy
Back Oxford
with Bk 2 and
Cambridge
with Bk 1
-1
-1
Q = 1.43 + 3.9
Q = 0.956 < 1
You can earn
4.6%
Bet 100 on Oxford with bookie 2 and 100 x 1.43 / 3.9 = 36.67 on Cambridge at bookie 1.
If Oxford win, you collect 100 x 1.43 = 143 from bookie 2.
If Cambridge win, you could collect 36.67 x 3.9 = 143 from bookie 1.
You invested 136.67 and collect 143, a profit of 6.33 (4.6%) no matter what the outcome.
What About the Q > 1 Situations
This is the money-laundering case
You are guaranteed a loss of (1 - 1/Q) of your stake money
That is the cost of the laundering and carries no risk of greater loss
Weird Judging
Means
Ice Skating
Ladies Figure Skating
Salt Lake City Olympics
Before the last competitor skates…
Skater
Short
Long
Total
Kwan
0.5
2.0
2.5
Hughes
2.0
1.0
3.0
Cohen
1.5
3.0
4.5
?
?
Slutskaya 1.0
Lowest scores lead
And after Slutskaya skates…
Skater
Short
Long
Total
Hughes
2.0
1.0
3.0
Slutskaya 1.0
2.0
3.0
Kwan
0.5
3.0
3.5
Cohen
1.5
4.0
5.5
Hughes wins by tie-break!
Slutskaya has changed the order of Hughes and Kwan
Holyfield vs Lewis (1999)
R
1
2
3
1
L
L
L
2
L
L
L
3
H
H
H
4
H
L
L
5
H
L
L
6
L
H
L
7
L
D
L
8
H
H
H
9
H
H
H
10
H
D
H
Judge 1: 7- 5 Holyfield
Judge 2: Draw 5- 5
Judge 3: 7- 4 Lewis
This is scored as a draw
Even though Lewis has won 17-16 on rounds
11
H
H
D
12
L
L
L
The Moral
Don’t add preferences or ranks
If A best B and B beat C
It doesn’t mean A beats C
Preference votes ABC, BCA, CAB
imply
A bts B 2-1 and B bts C 2-1
But
C bts A 2-1
The Three-Box Trick
Monty Hall – 2 goats and 1 car
You choose Box 1: he opens Box 3
1
Prob = 1/3
1
Prob = 1/3
2
3
Prob = 1/3
Prob = 1/3
 Prob = 2/3 
2
now open 3
Prob = 2/3
Prob = 0
So you should switch from Box 1 to Box 2
You are Twice As Likely to Win if You
Switch than if You Don’t !