Question

Sita and Gita both have a pair of tetrahedral dice, whose faces are marked as 1,2,3 and 4, the lowest face being regarded as the outcome. They throw their dice simultaneously. If $P=\frac{a}{b}$ be the probability expressed in lowest term, that the sum of the numbers on their dice is different, find the least value of $(a+b),a,b\in N$.

Answer 1

Answer: 117

$P=1$ - (Probability that they have equal throw)
$=1-\left[\left(\frac{1}{16}\times \frac{1}{16}\right)+\left(\frac{2}{16}\times \frac{2}{16}\right)+\left(\frac{3}{16}\times \frac{3}{16}\right)+\left(\frac{4}{16}\times \frac{4}{16}\right)+\left(\frac{3}{16}\times \frac{3}{16}\right)+\left(\frac{2}{16}\times \frac{2}{16}\right)+\left(\frac{1}{16}\times \frac{1}{16}\right)\right]$
$=\left[1-\frac{28+16}{16\cdot 16}\right]=\left[1-\frac{44}{16\cdot 16}\right]=\left[1-\frac{11}{64}\right]=\frac{53}{64}$
$\text{ Hence, }(a+b{)}_{\text{least }}=53+64=117$