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GuesstheBiggerNumber MathematicsforComputerScience MIT 6.042J/18.062J Team1: Writetwointegersfrom0to7ontwo Introductionto piecesofpaper ShowtoTeam2facedown RandomVariables Team2:

  1. Guess the Bigger Number Mathematics for Computer Science MIT 6.042J/18.062J Team 1: • Write two integers from 0 to 7 on two Introduction to pieces of paper • Show to Team 2 face down Random Variables Team 2: • Expose one paper and look at number Bigger Number Game • Either stick or switch to other number Team 2 wins if gets larger number Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.1 ranvarbigger.2 Guess the Bigger Number Guess the Bigger Number Do you think one team has an Do you think one team has an advantage? advantage? Which one? You might like to try playing the game a few times with some teammates before seeing the answers below. Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.3 ranvarbigger.4 1

  2. Strategy for Team 2 Analysis of Team 2 Strategy • pick a paper to expose, giving each Let low < high be the paper equal probability. • if exposed number is “small” then integers chosen by Team 1. switch, otherwise stick. That is There are three cases: switch if � threshold Z where Z is a random integer E [0,7) Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.5 ranvarbigger.6 Analysis of Team 2 Strategy Analysis of Team 2 Strategy Case M: low � Z < high Case H : high � Z Team 2 wins in this case, so Team 2 will switch, so wins iff low card gets exposed 1 Pr[Team 2 wins | M] = 1 1 Pr[Team 2 wins | H ] = and Pr[M] �� 2 7 Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.7 ranvarbigger.8 2

  3. Analysis of Team 2 Strategy Analysis of Team 2 Strategy So � 1/7 of time, sure win. Case L : Z < low Rest of time, win 1/2. Team 2 will stick, so wins iff high card gets exposed 1 By Law of Total Probability Pr[Team 2 wins | L ] = 2 Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.9 ranvarbigger.11 Analysis of Team 2 Strategy Analysis of Team 2 Strategy So � 1/7 of time, sure win. So � 1/7 of time, sure win. Rest of time, win 1/2. Rest of time, win 1/2. Pr[Team 2 wins] =� Pr[Team 2 wins] ��   4 ⋅ 1 + 1 ⋅ 1 − 1 Pr[win | M] · Pr[M] +�  = 1  __ __ 7 M ] · Pr[ ]   Pr[win | M 7 2 7 Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.12 ranvarbigger.14 3

  4. Analysis of Team 2 Strategy Analysis of Team 2 Strategy So Team 2 has the So Team 2 has the advantage advantage, no matter what Team 1 does! Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.15 ranvarbigger.16 Team 1 Strategy Optimal Strategy 4 …& Team 1 can play so Pr[Team 2 wins] =� 4 7 Pr[Team 2 wins] � 7 is optimal for both no matter what Albert R Meyer May 6, 2013 Albert R Meyer May 6, 2013 ranvarbigger.17 ranvarbigger.18 4

  5. Random Variables Informally: an RV is a number produced by a random process: • threshold variable Z • number of exposed card • number of larger card • number of smaller card Albert R Meyer May 6, 2013 ranvarindep.19 5

  6. MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 20 15 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

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