<s>[INST] Replacing 8 red balls with blue balls You have a bag with 8 red balls in it.
Every turn you randomly select a ball from the bag. If it is red you replace it with a blue ball, and if it is blue you replace it with a red ball.
What is the expected number of turns you need to make before all 8 balls in the bag are blue?
If the answer is $\frac{a}{b}$, where a and b are coprime positive integers, what is $a+b$?

My Approach:
$E_n$ is the expected number of moves given that you have $n$ blue balls
$E(7) = \frac{1}{8}(1) + \frac{7}{8}[1 + E(6)]$
$E(6) = \frac{2}{8}[1 + E(7)] + \frac{6}{8}[1 + E(5)]$
$E(5) = \frac{3}{8}[1 + E(6)] + \frac{5}{8}[1 + E(4)]$
$E(4) = \frac{4}{8}[1 + E(5)] + \frac{4}{8}[1 + E(3)]$
$E(3) = \frac{5}{8}[1 + E(4)] + \frac{3}{8}[1 + E(2)]$
$E(2) = \frac{6}{8}[1 + E(3)] + \frac{2}{8}[1 + E(1)]$
$E(1) = \frac{7}{8}[1 + E(2)] + \frac{1}{8}[1 + E(0)]$
$E(0) = \frac{8}{8}[1 + E(1)] $
But in this way it doesn't seem to go well.
 [/INST] I am not finding anything wrong in the equations.
Putting it in a Jordan- Gauss calculator, I get an answer of $\dfrac{a}{b}= \dfrac{32768}{105}$
Thus $a+b= 32873$
The calculator is here
I strongly advise you to check the computations.
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