E1 Sophie lives in New York City To go from point A to point

E1 Sophie lives in New York City. To go from point A to point B, she has to go through a confusing intersection, located one hour from point A. At this intersection, there are three roads one could take. ?f you take road 1, you get to point B in 1 hour. If you lake road 2, you get back to point A after .5 hour, and start over. If YOU take road 3, you return to point A alter 1/3 hour, and start over. Whenever Sophie gets to the intersection, there is someone tailgating her and honking, so she doesn?t think straight and always winds up picking the road completely at random. Find the expected value of the time it takes for Sophie to reach point B.

Solution

Sophie has 3 alternate roots

I is from A to B in road 1 in 1 hour

II is from A to A in 0.5 hours and again to B in 1 hour

III is from A to A in 1/3 hour and again to B in 1 hour.

That is whenever she comes to intersection she has confusion and hence has a prob of selecting correct road in 1/3 ways

Otherwise she takes road 2 in 1/3 and takes 1.5 hours

or road 3 and takes 1 1/3 hours

If second time also she does mistake again this recurs as a continuous process till she selects road 1

Imagin the scernio she came to intersection.

1/3 going to correct road or 1/3 road 2 or 1/3 road 3

If road 2 is taken again going back to A coming to intersection again same 3 possibilities

Similarly for road 3

Thus we can say either in 1 hour Reaching B or 1/3 (1/2+1/2^2+...) in road 2 and 1/3(1+1/3^2+1/3^3+....................) in road 3

Thus time taken are either 1 hour or 2/3 hours or 1/2 hours

Expected time = 1/3(1+2/3+1/2)

= 13/3 hours (apprxy)