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My transition matrix is $$\left( \begin{array}{ccc} .4 & .3 & .3 \\ .3 & .2 & .5 \\ .7 & 0 & .3 \end{array}\right) $$ So I'm asked for $P[X_3=1,\,X_2=1\,\lvert\,X_1=2]$ and: $$\begin{eqnarray}P[X_3=1,\,X_2=1\,\lvert\,X_1=2]&=&\dfrac{P[X_1=2,\,X_2=1\,,\,X_3=1]}{P[X_1=2]}\\ &=& \dfrac{\sum_{i=1}^3P[X_1=2,\,X_2=1\,,\,X_3=1\,\lvert\,X_0=i]P[X_0=i]}{\sum_{i=1}^3P[X_1=2\,\lvert\,X_0=i]P[X_0=i]}\\ &=&\dfrac{\sum_{i=1}^3P[X_0=i\,,\,X_1=2,\,X_2=1\,,\,X_3=1]}{\sum_{i=1}^3P[X_1=2\,\lvert\,X_0=i]P[X_0=i]}\\ &=&\ldots \end{eqnarray} $$ And the rest is easy to calculate. Am I right about the previous?

Erick
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    $$\mathbb P(X_3=1,X_2=1\mid X_1=2) = \mathbb P(X_3=1|X_2=1)\mathbb P(X_2=1\mid X_1=2)$$ and these probabilities may be computed by the Chapman-Kolmogorov equations. – Math1000 Sep 20 '16 at 02:31

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