probability proof requiring conditional probability 
Consider the experiment where two dice are thrown. Let $A$ be the event that the sum of the two dice is 7. For each $i\in\{1,2,3,4,5,6\}$, let $B_i$ be the event that at least one $i$ is thrown.
(a) Compute $P(A)$ and $P(A\mid B_1)$.
(b) Prove that $P(A\mid B_i)=P(A\mid B_j)$ for all $i$ and $j$.
(c) Since you know that some $B_j$ always occurs, does it make sense that $P(A)\neq P(A\mid B_i)$? (After all, if $E$ is an event with $P(E)=1$, then for any event $F$, $P(F\mid E)=P(F)$. What is going on? Does this seem paradoxical?)

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I am stuck on this question especially the proof. Can anybody help me out.

Hint:
          $\qquad\qquad\text{Event }A\\\begin{array}{c|c|c|c|c|c|c|c|}
 & 1 & 2 & 3 & 4 & 5 & 6\\\hline
1&  &   &   &   &   &  \bullet \\\hline
2&  &   &   &   & \bullet  &   \\\hline
3&  &   &   & \bullet  &   &   \\\hline
4&  &   & \bullet  &   &   &   \\\hline
5&  &  \bullet &   &   &   &   \\\hline
6& \bullet &   &   &   &   &   \\\hline
\end{array}$                $\qquad\qquad\text{Event }B_1\\\begin{array}{c|c|c|c|c|c|c|c|}
 & 1 & 2 & 3 & 4 & 5 & 6\\\hline
1& \bullet & \bullet  &  \bullet & \bullet  & \bullet  & \bullet  \\\hline
2& \bullet &   &   &   &   &   \\\hline
3& \bullet &   &   &   &   &   \\\hline
4& \bullet&   &   &   &   &   \\\hline
5& \bullet &   &   &   &   &   \\\hline
6& \bullet &   &   &   &   &   \\\hline
\end{array}$  
          $\qquad\qquad\text{Event }B_2\\\begin{array}{c|c|c|c|c|c|c|c|}
 & 1 & 2 & 3 & 4 & 5 & 6\\\hline
1&  &  \bullet &   &   &   &   \\\hline
2&\bullet  &  \bullet &   \bullet&  \bullet &  \bullet & \bullet  \\\hline
3&  &  \bullet &   &   &   &   \\\hline
4&  &  \bullet &   &   &   &   \\\hline
5&  &  \bullet &   &   &   &   \\\hline
6& &  \bullet &   &   &   &   \\\hline
\end{array}$                $\qquad\qquad\text{Event }B_3\\\begin{array}{c|c|c|c|c|c|c|c|}
 & 1 & 2 & 3 & 4 & 5 & 6\\\hline
1&  &   &  \bullet &   &   &   \\\hline
2&  &   & \bullet  &   &   &   \\\hline
3& \bullet &  \bullet & \bullet  &  \bullet & \bullet  & \bullet  \\\hline
4&  &   & \bullet  &   &   &   \\\hline
5&  &   & \bullet  &   &   &   \\\hline
6& &   & \bullet  &   &   &   \\\hline
\end{array}$  
          $\qquad\qquad\text{Event }B_4\\\begin{array}{c|c|c|c|c|c|c|c|}
 & 1 & 2 & 3 & 4 & 5 & 6\\\hline
1&  &   &   & \bullet  &   &   \\\hline
2&  &   &   & \bullet  &   &   \\\hline
3&  &   &   &  \bullet &   &   \\\hline
4& \bullet &  \bullet &  \bullet & \bullet  & \bullet  & \bullet  \\\hline
5&  &   &   & \bullet  &   &   \\\hline
6& &   &   &  \bullet &   &   \\\hline
\end{array}$                $\qquad\qquad\text{Event }B_5\\\begin{array}{c|c|c|c|c|c|c|c|}
 & 1 & 2 & 3 & 4 & 5 & 6\\\hline
1&  &   &   &   & \bullet  &   \\\hline
2&  &   &   &   & \bullet  &   \\\hline
3&  &   &   &   & \bullet  &   \\\hline
4&  &   &   &   & \bullet  &   \\\hline
5& \bullet &   \bullet&  \bullet &  \bullet & \bullet  & \bullet  \\\hline
6& &   &   &   &\bullet   &   \\\hline
\end{array}$  
          $\qquad\qquad\text{Event }B_6\\\begin{array}{c|c|c|c|c|c|c|c|}
 & 1 & 2 & 3 & 4 & 5 & 6\\\hline
1&  &   &   &   &   &   \bullet\\\hline
2&  &   &   &   &   &  \bullet \\\hline
3&  &   &   &   &   &  \bullet \\\hline
4&  &   &   &   &   &  \bullet \\\hline
5&  &   &   &   &   & \bullet  \\\hline
6& \bullet& \bullet  & \bullet  &\bullet   &  \bullet &  \bullet \\\hline
\end{array}$ 
