Two dice are thrown together. Find the probability that the sum of the numbers on their uppermost faces is a perfect square or divisible by 4.

The probability of an event E occurring is the ratio of the number of cases in its favour to the total number of equally likely cases.

$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{\text{ no. of cases favourable to event }E}{\text{ Total no. of cases }}$

Two fair dice thrown

so total cases $=6\times 6=36$

and perfect square or divisible by 4 that mean that sum should be equal to $=4,8,9,12$

for getting $4=(1,3) ;(2,2) ;(3,1)$

for getting $8=(2,6) ;(3,5) ;(4,4) ;(5,3) ;(6,2)$

for getting $9=(3,6) ;(4,5) ;(5,4) ;(6,3)$

for getting $12=(6,6)$

so total cases $=3+5+4+1=13$

so, probability $=\frac{13}{36}$