Paradox Lost: The Evils of Coins and Dice

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

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<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.

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,

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. 2 3 4 5 6 7 8 9 10 11 12 Probability Relative Probability Expected # Rolls Wins Race Wins Race 19.8 0.247 0.499 21.2 0.248 0.501

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

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

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

13. 2 3 4 5 6 7 8 9 10 11 12

14. 2 3 4 5 6 7 8 9 10 11 12

15. 2 3 4 5 6 7 8 9 10 11 12

16. Relative probability down to 0.278 from 0.293. 2 3 4 5 6 7 8 9 10 11 12

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

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

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. HTHH vs HHTT • Which happens fastest on average? • Which is more likely to win a race?

24. Expected Number of Flips to See the Sequence

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 S1,…,Sn • Probabilities of winning p1,…,pn • Expected number of flips E

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

29. Probabilities of Winning Races

30. Probabilities of Winning Races

31. Probabilities of Winning Races