Question

A bag contains 3 white chips and 3 red chips. you repeatedly draw a chip at random from the bag. if it's white, you set it aside; if it's red, you put it back in the bag. after removing all 3 white chips, you stop. what is the expected number of times you will draw from the bag?

Answer 1

we have to solve the following recursive relations.

Let h(k) be the expected number of steps until your reach the (absorbing) state 4 when you are in state k, for k=1,2,3,4.

So we have that h(4)=0

because when you are already in 4 you need zero steps to reach 4.

Then for k=3 h(3)=1+0.75h(3)+0.25h(4)
because when you are in state 3 you will do one step (+1) and you will reach with probability 0.75 again state 3 and with probability 0.25 state 4. And you start over (to count the expected number of steps) from the new state, therefore 0.25h(4) and 0.75h(3).

Similarly you can determine h(2) and h(1) and solve the system of equations to determine h(1) which is the expected value of steps to reach state 4 from the initial state which is 1 (according to your notation of the states).

More precisely:
h(2)=1+0.6h(2)+0.4h(3)
and
h(1)=1+0.5h(1)+0.5h(2)
which gives the following system
h(4)=1
h(3)=4+h(4)
h(2)=2.5+h(3)
h(1)=2+h(2)

which gives h(4)=1,h(3)=4,h(2)=6.5,h(1)=8.5.

So the expected number of steps is 8.5.