Question

After his bargain with Dormammu, Dr. Strange is trying to get back to Earth from the Dark Dimension. To this end, he manages to open three portals:

• Portal 1 can lead him directly back to Earth after 5 minutes.

• Portal 2 leads him through the Nine Realms and back to the Dark Dimension (where he started) after 15 minutes.

• Portal 3 leads him through the perilous Quantum Realm and back to the Dark Dimension after 25 minutes (assuming he survives the trip, of course).

Since Dr. Strange does not know which portal is the correct one to take and the portals change positions every time he goes through one of them, he decides to pick one portal (out of three) at random with equal probability. What is the expected time for Dr. Strange to get back to Earth?

Answer 1

Let x(in minutes) denote the expected time for Dr. Strange to get back to earth

In case of portal 2, he gets back to the dark dimension after 15 minutes. Thus, expected time to get back to earth from portal 2 = 15 + x

Expected time to get back to earth from portal 3 = 25 + x

Now, to choose each of the three portals is equi-probable with probability = 1/3

Thus, x = 1/3 { 5 + (15 + x) + (25 + x) }

-> x = 45

Thus, expected time = 45 minutes