Q:

Determining the probability of throwing $8$ with two dice

Suppose we have two dice.  Now we are imposing a condition that the two dice are independent of each other and  when we throw, we must have $8$ (that means $6+2 , 5+3$ or $4+4$ etc).  Therefore we  are throwing two dice at the same time.  

What is the probability that we  get 8 in each time? 

A:

If first dice gives $r,1\le r\le 6;$ the second needs to give $8-r$
So, $1\le 8-r\le 6\implies 7\ge r\ge 2$
So, $2\le r\le 6$
So, the required probability $\sum_{2\le r\le 6}P(r)P(8-r)=5\cdot\left(\frac16\right)^2$ 

