Paradox Lost:The Evils of Coins and Dice

George GilbertOctober 6, 2010

0.66667 0 1 2 3 4 5 6 7 8 9 100 0 3 6 9 12 15 18 21 24 27 30

1 1.5 3.50 6.17 9.06 12.02 15.01 18.00 21.00 24.00 27.00 30.00

2 3 4.33 6.56 9.22 12.09 15.03 18.01 21.00 24.00 27.00 30.00

3 4.5 5.39 7.17 9.54 12.24 15.10 18.04 21.02 24.01 27.00 30.00

4 6 6.59 7.98 10.02 12.50 15.23 18.11 21.05 24.02 27.01 30.00

5 7.5 7.90 8.95 10.66 12.88 15.45 18.22 21.10 24.05 27.02 30.01

6 9 9.26 10.05 11.46 13.41 15.77 18.40 21.20 24.10 27.05 30.02

7 10.5 10.68 11.26 12.39 14.07 16.20 18.67 21.36 24.19 27.09 30.05

8 12 12.12 12.55 13.44 14.86 16.76 19.03 21.58 24.32 27.17 30.09

9 13.5 13.58 13.89 14.59 15.77 17.43 19.50 21.89 24.51 27.28 30.15

10 15 15.05 15.28 15.82 16.79 18.21 20.07 22.28 24.77 27.44 30.25

• What’s Best? Arthur T. Benjamin and Matthew T. Fluet, American Mathematical Monthly 107:6 (2000), 560-562.

Definition: The qth percentile is the number k for which P (X<k)<q/100 and P(X≤k)>q/100. The 50th percentile is also called the median.

Theorem (Benjamin,Fluet) Flipping a coin with probability of heads p, the configuration of n coins which has minimal expected time to remove all n is the pth percentile of the binomial distribution with parameters n and p.

Proof. Flip the coin n times and let X be the number of heads.

€

E[k,n − k] − E[k −1,n − k +1] = P(X ≤ k −1)⋅ 1p

− P(X ≥ k)1

1 − p=P(X ≤ k −1)

p−

1 − P(X ≤ k −1)1 − p

< 0

⇔P(X ≤ k −1) < p

Theorem (Benjamin,Fluet) Flipping a coin with probability of heads p, the configuration of n coins that wins over half the time against any other configuration is the median of the binomial distribution with parameters n and p.

Illustration of proof (our case).Flip the coin n times. From the binomial distribution,

P(X<6) 0.350

P(X=6) 0.273

P(X=7) 0.234

P(X>7) 0.143

• The Best Way to Knock ’m Down, Art Benjamin and Matthew Fluet, UMAP Journal 20:1 (1999), 11-20.

• The River Crossing Game, David Goering and Dan Canada, Mathematics Magazine 80:1 (2007), 3-15.

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

Expected # Rolls Wins Race Wins Race

19.8 0.2470.499

21.2 0.2480.501

Relative ProbabilityProbability

2 3 4 5 6 7 8 9 10 11 12

Relative probability (and probability) wins race is 0.293.

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

Relative probability down to 0.278 from 0.293.

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

Relative probability increases to 0.517 by the time 28 ships are on 5 and ultimately to 1.

2 3 4 5 6 7 8 9 10 11 12

2 3 4 5 6 7 8 9 10 11 12

Relative probability is small and decreases at first, but ultimately increases to 1.

• Waiting Times for Patterns and a Method of Gambling Teams, Vladimir Pozdnyakov and Martin Kulldorff, American Mathematical Monthly 133:2 (2006), 134-143.

• A Martingale Approach to the Study of Occurrence of Sequence Patterns in Repeated Experiments, Shuo-Yen Robert Li, The Annals of Probability 8:6 (1980), 1171-1176.

HTHH vs HHTT

HTHH vs HHTT

• Which happens fastest on average?

• Which is more likely to win a race?

Expected Number of Flips to See the Sequence

30 HHHH20 HTHT18 HTHH HHTH HTTH16 HHTT HHHT HTTT

• The expected duration for sequences with more than two outcomes and not necessarily equal probabilities, e.g. a loaded die, is still

• For different sequences R and S, not necessarily of the same length,

still makes computational sense. S is the one sliding; order matters!

€

R⊗S€

S⊗S

• The expected time to hit a sequence S given a head start R (not necessarily all useful) is

€

S⊗S − R⊗S.

Racing sequences S1,…,Sn

• Probabilities of winning p1,…,pn

• Expected number of flips E

€

Sk ⊗ Sk = E + pi(i=1

n

∑ Sk ⊗ Sk − Si ⊗ Sk )

E − pi i=1

n

∑ Si ⊗ Sk = 0

Probabilities of Winning RacesHTHH 4/7

HHTT 3/7

# Flips 10.28…

Yet the expected number of flips to get HTHH is 18, versus 16 to get HHTT.

Probabilities of Winning RacesHTHH 4/7

HHTT 3/7

# Flips 10.28…

HHTT 9/16

THTH 7/16

# Flips 9.875

Probabilities of Winning RacesHTHH 4/7

HHTT 3/7

# Flips 10.28…

HHTT 9/16

THTH 7/16

# Flips 9.875

HTHH 5/14

THTH 9/14

# Flips 12.85…

Probabilities of Winning RacesHTHH 4/7

HHTT 3/7

# Flips 10.28…

HHTT 9/16

THTH 7/16

# Flips 9.875

HTHH 5/14

THTH 9/14

# Flips 12.85…

HTHH 1/4

HHTT 3/8

THTH 3/8

# Flips 8.25