Interesting probability question

Question

A particle transitions between three states, A, B, and C, as follows:

It proceeds from A to B with probability 0.7 and then to C with probability 0.3.

It always starts from B and ends at A.

It remains in C (the absorbing state).

Let the process begin in A. What is the predicted number of steps to achieve state C?

scratch games, thanks!

Answer 1

Fun problem! Here's my answer:

The number of steps may be expressed as an odd number from 1 to infinity based on the description of the problem. The process may end in one step if A -> C. It requires three steps if (A->B ->A)->C. Every time the particle goes from A to B it immediately goes back to A -- this may happen any number of times but it always takes two steps. Once it goes from A to C the particle stops and it can't get to C from B so (A->C) will always be the last step and adds one. If k is the number of times A makes a round trip through B then 2k is the number of steps taken and 2k+1 is the final number of steps.

The probability mass function for the steps may be expressed using a geometric random variable with success probability p=0.3 (the win condition) and (1-p)=0.7 (the keep playing condition). To get the expected number of steps we may use the formula:

E(steps) = Sum_(k=0 to infinity) of ( (2k+1) * (0.3) * (0.7)^k )

The first few terms look like this:

1 * 0.3

3 * 0.3 * 0.7

5 * 0.3 * 0.7^2

...

The sum converges to 5.66666... so on average this process will take 5 2/3 steps to complete.