Coin 1 comes up heads with probability $0.6$ and coin 2 with probability $0.5.$ A coin is continually flipped until it comes up tails, at which time that coin is put aside and we start flipping the other one. If we start the process with coin 1 what is the probability that coin 2 is used on the fifth flip? I put 0=Heads, 1= Tails My matrix is $P_{00}=0.6, P_{01}=0.4, P_{10}=0.5, P_{11}=0.5$. My answer is that the $P(\text{coin 2 on 5th flip}) = P^4$ (?) I know I need the 4th step transition but i'm not sure which subscript i need for my matrix.
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$\begingroup$I think you should make a clarification here: Let's denote $C_1$ the state where we use the unfair coin and $C_2$ when we flip the fair coin. Thus, the transition matrix will be something like that:
$$P = \begin{array}{l|ll} & C_1 & C_2 \\\hline C_1 & 0.6 & 0.4 \\ C_2 & 0.5 & 0.5 \end{array} \implies P^4 = \begin{bmatrix} 0.5556 & 0.4444\\ 0.5555 & 0.4445 \end{bmatrix} $$
$P^4_{12} = 0.4444 $ is the probability to reach $C_2$ in exactly 4 tosses when we start from $C_1,$ which means we will flip the second coin on the 5th toss.
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