Question

Two persons each make a single throw with a pair of dice. If $p$ denotes the probability that their throws are unequal, find $648p$

Answer 1

We find the probability of the complement event, that is, their throws are equal. There are $365$ outcomes for one roll of a pair of dice, so the total number of outcomes for the two persons making one throw each is $36\times 36=1296$. Now let $a_i$, with $2\le i\le 12$, be the number of ways to get a sum of $i$ showing on the pair of dice when they are rolled.
Then
$a_2=a_{12}=1,a_3=a_{11}=2,a_4=a_{10}=4,a_5=a_9=4,a_6=a_8=5,a_7=6$.
Each player can throw an i in $a_i$ ways, so both of them will throw an i in $a_i^2$ ways. Summing over all values of $i$, we see that the number of ways the throws of the two persons will be equal is
$a_2^2+a_3^2+...+a_{12}^2=2(1^2+2^2+3^2+4^2+5^2)+6^2$
$=\frac{2}{6}\left(5\right)\left(6\right)+6^2=146$
$\therefore 1-p=\frac{146}{1296}=\frac{73}{648}$
$648p=73$.