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Calm market after $2$ days

ProbabilityEasy

A market is either Calm or Stormy each day. From a Calm day, the next day is Calm with probability $0.9$ . From a Stormy day, the next day is Calm with probability $0.5$ . Today is Calm. What is the probability that the day after tomorrow is Calm? Give a decimal.

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Hint

Split by what tomorrow is, then use the law of total probability.

Solution

Tomorrow is Calm with probability $0.9$ and Stormy with probability $0.1$ .
Path Calm then Calm then Calm has probability $0.9 \times 0.9 = 0.81$ .
Path Calm then Stormy then Calm has probability $0.1 \times 0.5 = 0.05$ .
Adding the paths from these $2$ cases gives $0.81 + 0.05 = 0.86$ .
The final answer is $0.86$ .

Answer: $0.86$

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