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The chance of a clean run

StatisticsHard

A signal has a hidden state, on or off. Day $1$ is on with probability $0.5$ . Each state repeats the next day with probability $0.8$ . An up day has probability $0.75$ when on and $0.25$ when off, independently once the states are known. What is the probability that days $1$ and $2$ are both up days, whatever the hidden states were? Give a fraction in lowest terms.

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Hint

Keep the joint chance of each state and everything seen so far, day by day.

Solution

Day $1$ joint chances: $0.5 \times 0.75 = 0.375$ for on and $0.5 \times 0.25 = 0.125$ for off.
Move to day $2$ : on gets $0.375 \times 0.8 + 0.125 \times 0.2 = 0.325$ and off gets $0.375 \times 0.2 + 0.125 \times 0.8 = 0.175$ .
Weight by the second up day: $0.325 \times 0.75 = 0.24375$ and $0.175 \times 0.25 = 0.04375$ .
Add them: $0.24375 + 0.04375 = \frac{23}{80}$ .

Answer: $\frac{23}{80}$

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