50 Hard Probability Problems

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Here are two problems (#16 and #17), adapted  from the book Fifty Challenging Problems in Probability Frederick Mosteller, Dover Publications.

Problem 16:  Eight tennis players (call them A,B,C,D,E,G,F,H) are randomly assigned to start positions in a ladder tournament. Initially, position 1 plays position 2, position 3 plays 4,  5 plays 6 and 7 plays 8.  Second round has 2 matches: winner of (1,2) match plays winner of (3,4), and winner (5,6) plays winner(7,8). The winners of the two 2nd round matches play each other in the final match.   Player A wins against any of the others. Player B always beats any opponent except player A. What is the probability that player B wins the 2nd place trophy in the final match?

Problem 17: In the same tournament type as in the previous problem, assume that the 8 players are of equal skill levels so each has a 50-50 chance of winning any particular match. A and B happen to be twins. What is the probability that they will play each other in some match during the tournament?

I enjoy working on probability theory problems even though it's not my strongest subject.  The problems tend to be easy to state and understand, but sometimes irritatingly difficult to solve.    The "Fifty Challenging Problems" book has a number that are interesting, of which the two are presented here.   The problems in the book have solutions provided, but I only use them as the last resort.   Over the years I have developed the habit of experimentally finding  or verifying solutions to probability problems.  Modeling the problem in code is fun and confirms that the problem was understood.   The results of running a million trials takes only a second or two and, if both are done correctly,  ensures that experimental and analytical results will closely agree.

 I've used that approach here and the program presents both  my analytical solutions developed without reference to solutions in the book and the experimental results.  l promised myself to refer to the book solution  only if  my analytical and experimental result did not agree.  In this case they did, but that leaves 48 mores opportunities to fail and learn. 

I plan to augment this program with additional problems in the future so having this page as a base will help get over the documentation hurdle when I do.  

Addendum February 7, 2012:  Here are two more problems which I fond interesting and added today in Version 4:

Problem 18: If 100 coins are tossed, what is the probability that exactly 50 heads will be showing.?

Problem 19: Samuel Pepys wrote Isaac Newton to ask which of these events is more likely: that a person get (a) at least 1 six when 6 dice are rolled. (b) at least 2 sixes when 12 dice are rolled, or (c) at least 3
sixes when 18 dice are rolled.  What is the answer?

February 27, 2012:  Version 5.- adds one more problem:  

Problem 20: A, B, and C are to fight a three cornered pistol duel. All know that A's chance of hitting his
target is 0.3, C's is 0.5 and B never misses. They are to fire at their choice of target in
succession in order A, B, C, cyclically (but a hit man is out of the contest, neither shooter
nor shot at), until only one man remains. What should A's strategy be?

The analytical solution for this one involvesl geometric progressions because the chance that you will win at round N+1 is the chance that you had of winning at round N times the chance that both you and your opponent miss on your next turns (otherwise the match is over and there is no round N+1).  By definition, a sequence where each term is a constant multiple of the previous term is  a "geometric sequence".    It took a long time to find the correct analytical solution  because there are lots of chances for arithmetic errors in determining the initial value and the ratio  for the sequences.   Fortunately, once the appropriate initial value, a, and ratio, r, have been determined, the overall probability of winning for each contestant is the sum of the probabilities of the individual outcome which is a/(1-r).       

 Running/Exploring the Program 

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Created January 26,2012

Modified July 29, 2017



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