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

$P = 1$ - (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] $
$=\left[1-\frac{28+16}{16 \cdot 16}\right]=\left[1-\frac{44}{16 \cdot 16}\right]=\left[1-\frac{11}{64}\right]=\frac{53}{64}$
$\text { Hence, }(a+b)_{\text {least }}=53+64=117$