Hard | Probability |

An ant is standing on one corner of a cube & can only walk on the edges. The ant is drunk and from any corner, it moves randomly by choosing any edge! What is the expected number of edges the ant travels, to reach the opposite corner?

Hint

Try to find the equivalent vertices with respect to distance yet to travel. This should give 4 equivalent merged vertices, with 1st being start & 4th being destination.

Solution

Let the expected number of step required to go from (0,0,0) to (1,1,1) be E0.Also let expected number of step required to reach (1,1,1) from (0,0,1)(Or from (0,1,0) or from (1,0,0)) be E1. similarly expected number of step required to reach (1,1,1) from (0,1,1) (Or from (1,0,1) or from (1,1,0)) be E2.Then we can write:

E0=1/3(E1+E1+E1)+1

E1= 1/3*E0 + 2/3*E2 + 1

E2 = 2/3*E1 + 1

solving this we find E0 as 10.

E0=1/3(E1+E1+E1)+1

E1= 1/3*E0 + 2/3*E2 + 1

E2 = 2/3*E1 + 1

solving this we find E0 as 10.

Source: Quant Interview

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