- How to Be A Winner - - How to Be A Winner - The Maths of Race Fixing The Maths of Race Fixing

and and Money LaunderingMoney Laundering

John D Barrow

Why is Probability Theory Not Why is Probability Theory Not Ancient?Ancient?

Religious beliefsReligious beliefs

Or Or

No concept of No concept of equallyequally

likely outcomeslikely 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 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”nonexistent but are hidden, and we attribute them to the will of the true God”

Sheep's’ ankle bones, 6-sided, numbered, asymmetricalDivination with sets of 5 in Asia Minor from 3600 BC

Eventually replaced by dice

Astragali

Ancient Dice

The most popular dice game of theThe most popular dice game of the Middle Ages: was called “hazard”Middle Ages: was called “hazard”Arabic “al zhar” means “a die.”Arabic “al zhar” means “a die.”

Roman icosahedral die20 faces

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.

Right and Left-handed Dice

The Problem of the PointsChevalier 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 5/6Probability of no 6 in four throws is 5/6 5/6 5/6 5/6 = (5/6)4 = 625/1296Probability 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 = 0.5086Probability of one double six is 1 - (35/36)24 = 0.4914 < 1/2

After a while he stopped doing this !

Winning The TossWinning The TossAustralian Open January 2008

Playing Fair With a Biased Playing Fair With a Biased CoinCoin

Unequal probability of H and T: Unequal probability of H and T: p p ½ ½ Probability of H is Probability of H is ppProbability of T is Probability of T is 1-p1-pToss twice and ignore pairs Toss twice and ignore pairs HHHH and and TTTTProbability of HT is Probability of HT is p(1-p)p(1-p)Probability of TH is Probability of TH is (1-p)p(1-p)pCall combination HT ‘Newheads’Call combination HT ‘Newheads’Call combination TH ‘Newtails’Call combination TH ‘Newtails’Newheads and Newtails are equally likelyNewheads and Newtails are equally likelyEfficiency is poor (50%) – discard the Efficiency is poor (50%) – discard the HHHH and and TTTT s s

Faking Random SequencesFaking Random Sequences1.1. THHTHTHTHTHTHTHTHTTTHTHTHTHTHTHHTHHTHTHTHTHTHTHTHTTTHTHTHTHTHTHH

2.2. THHTHTHTHHTHTHHHTTHHTHTTHHHTHTTTTHHTHTHTHHTHTHHHTTHHTHTTHHHTHTTT

3.3. HTHHTHTTTHTHTHTHHTHTTTHHTHTHTHTTHTHHTHTTTHTHTHTHHTHTTTHHTHTHTHTT

Do these look like real random sequences ?

4. THHHHTTTTHTTHHHHTTHTHHTTHTTHTHHH5. HTTTTHHHTHTTHHHHTTTHTTTTHHTTTTTH6. TTHTTHHTHTTTTTHTTHHTTHTTTTTTTTHH

Some More CandidatesWith 32 tosses

Are they random?

4. THHHHTTTTHTTHHHHTTHTHHTTHTTHTHHH5. HTTTTHHHTHTTHHHHTTTHTTTTHHTTTTTH6. TTHTTHHTHTTTTTHTTHHTTHHHHHHTTTTH

Some More CandidatesWith 32 tosses

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) Winning (and Losing) StreaksStreaks

The Nasser Hussain EffectEngland cricket captain

During 2000-2001Atherton 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 tossesBut he captained England 101 times and there is a chance of about

1 in 180 of a losing streak of 14

“Flipping useless, Nasser!” BBC

Can You Always Win?

Or avoid ever losing ?

The Win-Win ScenarioThe Win-Win ScenarioThe odds for the runners are aThe odds for the runners are a11 to 1, a to 1, a22 to 1, a to 1, a33 to 1, and so on, for any number of runners to 1, and so on, for any number of runners in the race. in the race. If the odds are 5 to 4 then we express that as an aIf the odds are 5 to 4 then we express that as an a ii of 5/4 to 1 of 5/4 to 1 Bet a fraction 1/(aBet a fraction 1/(aii +1) of the total stake money on the runner with odds of a +1) of the total stake money on the runner with odds of a ii to 1 to 1If there are N runners, we will always make a profit ifIf there are N runners, we will always make a profit if

Q = 1/(aQ = 1/(a11 +1) + 1/(a +1) + 1/(a22 +1) + 1/(a +1) + 1/(a33 +1) +….+ 1/(a +1) +….+ 1/(aNN +1) < 1 +1) < 1

Winnings = (1/Q – 1) Winnings = (1/Q – 1) our total stake our total stake

Example:Example:Four runners and the odds for each are 6 to 1, 7 to 2, 2 to 1, and 8 to 1 and. 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 aThen we have a11 = 6, a = 6, a22 = 7/2, a = 7/2, a33 = 2 and a = 2 and a44 = 8 and = 8 andQ = 1/7 + 2/9 + 1/3 + 1/9 = 51/63 < 1Q = 1/7 + 2/9 + 1/3 + 1/9 = 51/63 < 1Allocate our stake money with 1/7 on runner 1, 2/9 on runner 2, 1/3 on runner 3, 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 and 1/9 on runner 4 We will win at least 12/51 of the money we staked (and of course we get our We will win at least 12/51 of the money we staked (and of course we get our stake money back as well).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/(aQ = 1/(a11 +1) + 1/(a +1) + 1/(a22 +1) +….. > 1 +1) +….. > 1

But if you know the favourite has been hobbled then you calculate Q excluding a1 which can result in

QQfixfix = 1/(a = 1/(a22 +1) + 1/(a +1) + 1/(a33 +1) + …. < 1 +1) + …. < 1

If there are 4 runners with odds3 to 1, 7 to 1, 3 to 2, and 1 to 13 to 1, 7 to 1, 3 to 2, and 1 to 1

Q = 1/4 + 1/8 + 2/5 + 1/2 = 51/40 > 1So we can’t guarantee a winning return

Dope the favourite and place you money on the otherthree 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 withQfix = 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

OutcomeOutcome 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 andCambridge with Bk 1

Q = 1.43-1 + 3.9-1

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.

When Bookies Disagree

What About the Q > 1 Situations

This is the money-laundering caseYou 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 SkatingLadies Figure Skating

Salt Lake City Olympics

SkaterSkater ShortShort LongLong TotalTotalKwanKwan 0.50.5 2.02.0 2.52.5

HughesHughes 2.02.0 1.01.0 3.03.0

CohenCohen 1.51.5 3.03.0 4.54.5

SlutskayaSlutskaya 1.01.0 ?? ??

Before the last competitor skates…

Lowest scores lead

SkaterSkater ShortShort LongLong TotalTotalHughesHughes 2.02.0 1.01.0 3.03.0

SlutskayaSlutskaya 1.01.0 2.02.0 3.03.0

KwanKwan 0.50.5 3.03.0 3.53.5

CohenCohen 1.51.5 4.04.0 5.55.5

And after Slutskaya skates…

Hughes wins by tie-break!Slutskaya has changed the order of Hughes and Kwan

Holyfield vs Lewis (1999)R 1 2 3 4 5 6 7 8 9 10 11 121 L L H H H L L H H H H L2 L L H L L H D H H D H L3 L L H L L L L H H H D L

Judge 1: 7-5 HolyfieldJudge 2: Draw 5-5Judge 3: 7-4 Lewis

This is scored as a draw Even though Lewis has won 17-16 on rounds

The MoralThe 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, CABimply

A bts B 2-1 and B bts C 2-1But

C bts A 2-1

The Three-Box TrickThe Three-Box Trick

Monty Hall – 2 goats and 1 car

1 2 3

Prob = 1/3 Prob = 1/3 Prob = 1/3 Prob = 2/3

1 2 now

open 3Prob = 1/3 Prob = 2/3

Prob = 0

You choose Box 1: he opens Box 3

So you should switch from Box 1 to Box 2

You are Twice As Likely to You are Twice As Likely to Win if You Switch than if Win if You Switch than if

You Don’t !You Don’t !