Mood forecast two days ahead

Question

Maya's mood each day is either cheerful or grumpy, and it depends only on how she felt the day before: - If she is cheerful today, she is cheerful tomorrow with probability 0.7 (and grumpy with probability 0.3). - If she is grumpy today, she is cheerful tomorrow with probability 0.5 (and grumpy with probability 0.5). She is cheerful on Monday. What is the probability that she is cheerful on Wednesday, two days later?

Accepted Answer

Wednesday is two steps after Monday, so we list the two-day routes that end in "cheerful" and add up their chances. Starting cheerful on Monday, Tuesday can be either mood, and then Wednesday must land on cheerful. - Cheerful Tuesday, then cheerful Wednesday: 0.7 	imes 0.7 = 0.49. - Grumpy Tuesday, then cheerful Wednesday: 0.3 	imes 0.5 = 0.15. These are the only two ways to be cheerful on Wednesday, and they cannot both happen, so we add them: 0.49 + 0.15 = 0.64. So there is a 64\% chance Maya is cheerful on Wednesday. Takeaway. To forecast two steps ahead in a Markov chain, split on the m…

Mood forecast two days ahead

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