Question

Two persons each make a throw with a pair of cubical die and let p denote the probability that their throws are equal. Find the value of 648p

Answer 1

The total number of cases $=6^4=1296$
Let the sum of the throws of the dice of the first person be $a$ and the second person be $b$.
Now, let us consider the number of cases when $a$ and $b$ acquire different values.
$a=b=2$ , Cases $=1\times 1=1$
$a=b=3$ , Cases $=2\times 2=4$
$a=b=4$ , Cases $=3\times 3=9$
$a=b=5$ , Cases $=4\times 4=16$
$a=b=6$ , Cases $=5\times 5=25$
$a=b=7$ , Cases $=6\times 6=36$
$a=b=8$ , Cases $=5\times 5=25$
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.
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$a=b=12$ , Cases $=1\times 1=1$
Total number of cases $=1+4+9+...+36+25+...+4+1=146$
Probability $=\frac{146}{1296}=\frac{73}{648}$
Hence, $648p=73$