# Probability in dice rolls

$A$ 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.

• you can easily get an approximation of the result using the CLT. Is it enough? – mookid Oct 27 '14 at 14:09
• @mookid But 20 and 21 aren't very large, are they? Can CLT be applied effectively? – pkwssis Oct 27 '14 at 14:12
• 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. – mookid Oct 27 '14 at 14:14
• @mookid Can this approach yield something more refined than the probability being somewhere around 50%? – Did Nov 5 '14 at 9:19

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 $0.6077$
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 $(-X + 7) \sim X.$