A pair of fair six-faced dice is thrown60 times. On each time, you bet one dolla

Question

A pair of fair six-faced dice is thrown60 times. On each time, you bet one dollar. Thereare

two options to choosefrom:

i) You win 29 dollars if the sum of thetwo dice is 2, and lose one dollar otherwise;

ii) You win 4 dollars if the sum of thetwo dice is 7, and lose one dollar otherwise;

   a) Which option has ahigher expected net-gain? Or are they the same?

   b) Which option has ahigher chance of coming out ahead (winning somemoney)?

(Hint: compare standard errors. You don’tneed to actually find the chances.)

Explanation / Answer

a) the probability of getting 2 on 2 dice is 1/36 so your expectedgain is 29(1/36)+(-1)(35/36)=-(1/6) while getting a six there are 6 available rolls {(1,6), (2,5),(3,4)} or you can swap them so the expected gain on the secondoption is 4(6/36)+(-1)(30/36)=-(1/6) which is exactly the samevalue therefore they are the same. b) In order to roll a 2 it is supposed to require 36 rolls whichmeans you would lose 7 dollars after you gain your 29. So in orderto gain money you have to roll a 2 by your 28th try and you wouldmake 1 dollar. However, in order to roll a 7, it only requires 6 rolls which wouldput you on -2 dollars every time you gain 4 dollars. So to make 1dollar you need a 7 by your 3rd roll. Thus you have (35/36)27 (1/6) = .0779 for option one and(35/36)2 (1/6) = .1575 for option two Therefore option two has a higher change of coming out ahead

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