A traveller is lost and at a point where three roads meet. One of the roads will bring her back to the same point after 1 hr of walking, another road will bring her back to the same point after 2 hrs of walking, and the third road will lead her home after 3 hrs of walking. Suppose the traveller uses the following strategy: she picks a road uniformly at random from all the roads she hasn’t yet tried. So, initially she picks one of the three roads with probability 1/3 , and if that road leads back to the original point, she picks one of the remaining ones with probability 1/2 each. What is the expected time it will take her to get home?
I was wondering what the best way would be to start off this question, as that is what I am mainly struggling with as I am quite new to the Expected Value Concepts.
Expected value is every outcome multiplied by how likely it is all added together. All the outcomes are this:
Go straight home (3 hr)
Try the 1 Hr path and then go home (4 hr)
Try the 2 Hr path and then go home (5 hr)
Try both paths before going home (6 hr)
The hardest part is deciding how likely each is. There aren't many possibilities, so a branching diagram is helpful. Three branches at first (each 1/3 chance) and from two of those branches another two branches (each with a local 1/2 chance). Multiply the total branch probabilities for a final probability. The first single branch has a 1/3 chance and all other double-branches have a 1/6 chance. Multiply by the hours:
3*1/3 + 4*1/6 + 5*1/6 + 6*(1/6+1/6) (The 6hr possibility happens on two of the branches) I get 4.5 hours like this.