On time the day after tomorrow

Question

A commuter bus is either on time or late each day, and tomorrow's status depends only on today's: - If it is on time today, it is on time tomorrow with probability 0.8 (and late with probability 0.2). - If it is late today, it is on time tomorrow with probability 0.4 (and late with probability 0.6). The bus is late today. What is the probability that the bus is on time the day after tomorrow, two days from now?

Accepted Answer

The day after tomorrow is two steps away. We branch on tomorrow's status and keep the routes that end "on time." We start late, tomorrow can be either status, and the final day must be on time. - Late, then late, then on time: 0.6 	imes 0.4 = 0.24. - Late, then on time, then on time: 0.4 	imes 0.8 = 0.32. Add the two routes that finish on time: 0.24 + 0.32 = 0.56. So the bus is on time two days from now with probability 0.56. Takeaway. A "two days later" Markov question is solved by summing over the intermediate day: list the routes, multiply the daily chances, and total the routes that rea…

On time the day after tomorrow

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