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Liquidity $2$ weeks ahead

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A dealer treats each week's liquidity as deep or thin, and the label is never published. A deep week is followed by a deep week with probability $0.7$ . A thin week is followed by a thin week with probability $0.6$ . This week is deep with probability $0.5$ . What is the probability that the week $2$ weeks from now is deep? Give a decimal.

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Hint

Push the belief through the transition table once per week.

Solution

After $1$ week, deep has probability $0.5 \times 0.7 + 0.5 \times 0.4 = 0.55$ , since a thin week turns deep with probability $0.4$ .
Repeat from $0.55$ : $0.55 \times 0.7 + 0.45 \times 0.4 = 0.565$ .

Answer: $0.565$

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