A Hunter has two dogs Intro to Probability by Dimitri Bertseka, Ch 1.5 Problem 30

Question

Here's the problem

A hunter has two hunting dogs. One day, on the trial of some animal, the hunter comes to a place where the road diverges into two paths. He knows that each dog, independent of the other, will choose the correct path with probability p. The hunter decides to let each dog choose a path and if they agree, take that one, and if they disagree, to randomly pick a path. Is his strategy better than just letting one of the two dogs decide on a path?

I started out by assigning event A to the case where the dog chooses the right path and made two tree-and-leaf diagrams for their decision (A or $A^c$).

I made a table for

$$\begin{array}{c|c|c|} \text{Outcome}& \text{Probability} & \text{Choice} \\ \hline \ D_1 = A \cap D_2 = A & p^2 & \text{Path A} \\ \hline \ D_1 = A \cap D_2 = A^c & p(1-p) & \text{Random} \\ \hline \ D_1 = A^c \cap D_2 = A & p(1-p) & \text{Random} \\ \hline \ D_1 = A^c \cap D_2 = A^c& (1-p)^2 & \text{Path B} \\ \hline \end{array}$$

I'm stuck here. I know what the answer is because it's available online. I can link it, if that's helpful. I'm just not sure how to progress in a way that gets the answer.

Additionally - I'm new to doing word problems like these. I'd be grateful if you could point me to guides or pointers for effectively breaking down these types of problems.

Answer 1

Your table is good.

Now then, the hunter selects the correct path if either

1. the dogs both select the correct path, or
2. the dogs select different paths and the Random selection, fair coin toss, choses the correct path.

What is the probability for the union of those events?   Is this better or worse than $p$?