Policy update equation - purpose of summation On the policy improvement equation, I'm confused about the need for the summation sign.

[Apologies for taking a picture - I didn't know how to type this equation in Latex
In the diagram below, if we wanted to estimate the policy for state s (left),my understanding is that we would compute each of:
pA * (rA + y* vA)
pB * (rB + y* vB)
pC * (rC + y* vC)
and whichever of the three (A,B,C) gives us the highest value will be the action we choose to take.
Why then do we have have a summation sign in the policy update equation? How would summing those three equations help?

 A: You haven't quite understood how all this works, because (or so it appears to me) your notation mixed up actions and states.
Let us consider the value of being in state $s$ at some arbitrary stage.  If we have actions $a \in \{A,B,C\}$ to choose from at that stage - state combination, we'll choose whichever action gives us the maximum expected reward, hence the term $\arg \max_a$ before the summation term.  Now, the reward of a particular action comes in two parts, the immediate reward $r(s,a)$ and the value of finding ourselves in some state $s'$ at the next stage, $V(s')$.  
Let's deal with the second term first.  When we select a particular action, in the general case there's no guarantee that we will end up at a particular state after the transition; it may be that action $A$ gives us probabilities $\{0.5,0.3,0.2\}$ of ending up in states $1,2,3$ respectively, action $B$ gives us probabilities $\{0.2,0.5,0.3\}$ for ending up in the three states respectively, and so forth.  Consequently, we need to calculate the expected reward related to the state transition by summing across all possible $V(s')$ weighted by the probabilities of finding ourselves in the various states $s'$ given the current state - action pair.  Since those values are dependent upon the policy $\Pi$ we're using, they are actually denoted $V_{\Pi}(s')$. The probabilities of the various future states $s'$ given the current state - action pair $(s,a)$ are denoted by $p(s'|s,a)$, so the expected value associated with the state transition can be written as:
$$\sum_{s'}p(s'|s,a)V_{\Pi}(s')$$
which is usually multiplied by a discount factor $\gamma$ to reflect the fact that future rewards are often considered less valuable the farther into the future they are.
Now for the first term.  Just because we've selected a particular action $a$ in state $s$ doesn't mean we have a guaranteed immediate reward $r$; the reward can be random, with a probability distribution depending on $s$ and $a$.  An obvious example is picking heads or tails in a coin flip; I win if I pick correctly, but I'm not guaranteed of winning, the coin flip itself affects the results.  So we can't just use the immediate reward, we have to calculate its expected value:
$$\sum_rp(r|s,a)r$$
Since both probabilities - of $r$ and $s'$ - are conditional upon only $s$ and $a$, we can combine the two expressions as follows:
$$\sum_{r,s'}p(r,s'|s,a)[r + \gamma V_{\Pi}(s')]$$
This gives us the expected value of choosing action $a$ when in  state $s$.  Our optimal policy $\Pi'$ when in state $s$ will choose the action that maximizes this value:
$$\Pi'(s) = \arg \max_a \sum_{r,s'}p(r,s'|s,a)[r + \gamma V_{\Pi}(s')]$$
