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$A$ rolls a standard $6$ sided die $20$ times while $B$ rolls it $21$ times. Find the probability that $A$'s outcome is more than $B$'s. Here, outcome means the sum of the numbers appearing on all the rolls.

I tried to approach this in many ways, using mostly generating functions but the cases were just too many to handle.

I even get a hint of the multinomial distribution of the $20$ and $21$ rolls, but I didn't go far with that too.

Please help me out. Hints and answers appreciated. Thank you.

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  • $\begingroup$ you can easily get an approximation of the result using the CLT. Is it enough? $\endgroup$ – mookid Oct 27 '14 at 14:09
  • $\begingroup$ @mookid But 20 and 21 aren't very large, are they? Can CLT be applied effectively? $\endgroup$ – pkwssis Oct 27 '14 at 14:12
  • $\begingroup$ they are not. But you look at an event in the central region, where it is the most powerful, I think it should be fine. $\endgroup$ – mookid Oct 27 '14 at 14:14
  • $\begingroup$ @mookid Can this approach yield something more refined than the probability being somewhere around 50%? $\endgroup$ – Did Nov 5 '14 at 9:19
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I just did it in Excel. Make rows for sums and columns for numbers of dice. In the column for 1 die, put $1/6$ in the first six cells as the chance of having that total. Then each cell in a later column is the sum of six cells in the column to its left divided by $6$. Copy right and down. A final column sums up the chance that B wins. I get $0.6077$

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  • $\begingroup$ I also did this, but arrived at 0.3577 not 0.3923 (note that the question is for A wins, 0.3923=1-0.6077). Monte Carlo simulation also gives a result closer to 0.3577. I also wonder whether this method is in the spirit of the question, "but the cases were just too many to handle". An excel or software approach of course makes it possible to handle more cases. But, this is a very nice, visual approach that shows exactly what is going on with the probability distribution of the different sums at each roll. Nice! $\endgroup$ – Anton Nov 5 '14 at 10:43
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Let $X_A$ be the total on $A$'s dice, $X_B$ be the total on $B$'s dice. Obviously $X_A + X_B$ is a multinomial variable; but $X_A - X_B + k$ is also a multinomial variable for some suitable choice of constant $k$. Find $k$, and then you merely need to find the probability that a certain multinomial will exceed a certain value.

Further hint: If $X$ has uniform distribution over the integers $1$ through $6,$ then $-X$ has uniform distribution over the integers $-6$ through $-1,$ and $(-X + 7) \sim X.$

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