How to Be A Winner

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- How to Be A Winner - The Maths of Race Fixing
and Money Laundering
John D Barrow
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Why is Probability Theory Not Ancient?

• Religious beliefs
• Or
• No concept of equally
• likely outcomes
• ? ? ?

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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
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Astragali
Sheep's ankle bones, 6-sided, numbered,
asymmetrical Divination with sets of 5 in Asia
Minor from 3600 BC Eventually replaced by dice
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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
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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.
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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
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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 gt 1/2
So he thought that he should bet on one or more
double 6s 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 lt 1/2
After a while he stopped doing this !
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Winning The Toss Australian Open January 2008
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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

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Faking Random Sequences

1. THHTHTHTHTHTHTHTHTTTHTHTHTHTHTHH
2. THHTHTHTHHTHTHHHTTHHTHTTHHHTHTTT
3. HTHHTHTTTHTHTHTHHTHTTTHHTHTHTHTT

Do these look like real random sequences ?
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Some More Candidates With 32 tosses
4. THHHHTTTTHTTHHHHTTHTHHTTHTTHTHHH 5.
HTTTTHHHTHTTHHHHTTTHTTTTHHTTTTTH 6.
TTHTTHHTHTTTTTHTTHHTTHTTTTTTTTHH
Are they random?
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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 gt 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
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Winning (and Losing) Streaks
The Nasser Hussain Effect England cricket
captain During 2000-2001 Atherton took over for
one game after he had lost 7 and won the toss
Normal service was then resumed
Flipping useless, Nasser! BBC
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
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Can You Always Win?
Or avoid ever losing ?
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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) lt 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 lt 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).

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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 gt 1 with all runners included Q
1/(a1 1) 1/(a2 1) .. gt 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) . lt 1
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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 gt 1 So we cant 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
lt 1 Whatever the outcome you will never do worse
than winning your stake money plus (40/31)
-1 ? Stake money 9/31 ? Stake money
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When Bookies Disagree
Outcome Bookmaker 1s odds Bookmaker 2s odds
Oxford win 1.25 1.43
Cambridge win 3.9 2.85
Q of Bookie 1 1.056 gt1 He gains 5.6
Q of Bookie 2 He gains 5.1 1.051 gt 1
A Mixed Strategy
Back Oxford with Bk 2 and Cambridge with Bk 1 Q 1.43-1 3.9-1 Q 0.956 lt 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.
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What About the Q gt 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
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Weird Judging Means Ice Skating
Ladies Figure Skating Salt Lake City Olympics
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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
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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
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This is scored as a draw Even though Lewis has
won 17-16 on rounds
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The Moral
Dont add preferences or ranks If A best B and B
beat C It doesnt 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
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The Three-Box Trick
Monty Hall 2 goats and 1 car
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You choose Box 1 he opens Box 3
1 2 3
Prob 1/3 Prob 1/3
Prob 1/3 ?
Prob 2/3 ?
1 2 now open 3 Prob 1/3
Prob 2/3 Prob 0
So you should switch from Box 1 to Box 2
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You are Twice As Likely to Win if You Switch than
if You Dont !
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