1. Paradox Lost: The Evils of Coins and Dice George Gilbert October 6, 2010

2. 0.66667 012345678910 0 0369121518212427 30 1 1.53.506.179.0612.0215.0118.0021.0024.00 27.0030.00 2 34.336.569.2212.0915.0318.0121.00 24.0027.0030.00 3 4.55.397.179.5412.2415.1018.04 21.0224.0127.0030.00 4 66.597.9810.0212.5015.23 18.1121.0524.0227.0130.00 5 7.57.908.9510.6612.88 15.4518.2221.1024.0527.0230.01 6 99.2610.0511.46 13.4115.7718.4021.2024.1027.0530.02 7 10.510.6811.26 12.3914.0716.2018.6721.3624.1927.0930.05 8 1212.12 12.5513.4414.8616.7619.0321.5824.3227.1730.09 913.5 13.5813.8914.5915.7717.4319.5021.8924.5127.2830.15 10 1515.0515.2815.8216.7918.2120.0722.2824.7727.4430.25

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

4. Definition: The qth percentile is the number k for which P (X q/100. The 50 th 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.

5. 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

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

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

8. 2 3 4 5 6 7 8 9 10 11 12

9. Expected # Rolls Wins Race Wins Race 19.80.2470.499 21.20.2480.501 Relative ProbabilityProbability

10. 2 3 4 5 6 7 8 9 10 11 12 Relative probability (and probability) wins race is 0.293.

11. 2 3 4 5 6 7 8 9 10 11 12

16. Relative probability down to 0.278 from 0.293.

17. 2 3 4 5 6 7 8 9 10 11 12

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

19. 2 3 4 5 6 7 8 9 10 11 12

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

21. 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.

22. HTHH vs HHTT

23. Which happens fastest on average? Which is more likely to win a race?

24. Expected Number of Flips to See the Sequence 30HHHH 20HTHT 18HTHH HHTH HTTH 16HHTT HHHT HTTT

25. 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!

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

27. Racing sequences S 1,…,S n Probabilities of winning p 1,…,p n Expected number of flips E

28. Probabilities of Winning Races HTHH4/7 HHTT3/7 # Flips10.28… Yet the expected number of flips to get HTHH is 18, versus 16 to get HHTT.

29. Probabilities of Winning Races HTHH4/7 HHTT3/7 # Flips10.28… HHTT9/16 THTH7/16 # Flips9.875

30. Probabilities of Winning Races HTHH4/7 HHTT3/7 # Flips10.28… HHTT9/16 THTH7/16 # Flips9.875 HTHH5/14 THTH9/14 # Flips12.85…

31. Probabilities of Winning Races HTHH4/7 HHTT3/7 # Flips10.28… HHTT9/16 THTH7/16 # Flips9.875 HTHH5/14 THTH9/14 # Flips12.85… HTHH1/4 HHTT3/8 THTH3/8 # Flips8.25