In a far away and long ago, there are only two weather states, rain and sun....

In a far away and long ago, there are only two weather states, rain and sun. If it's sunny today the probability it's be sunny tomorrow is 0.8. If it's rainy today the probability it'll be sunny tomorrow is 0.4. Weather changes are well described by a Markov chain. a) It's sunny today, and tomorrow is the start a 4-day holiday. What is the probability that all 4 days are sunny? b) A parade scheduled for the last day of the holiday. What is the probability of rain on the parade? c) A royal wedding is planned for one year (365) from today. If it's an outdoor wedding what is the probability the wedding gets rained out.

Hint: This is the steps. Thank you

n a land far away and long ago, there are only two weather states, rain and sun. If it's sunny today the probability it'll be sunny tomorrow is 0.8. If it's rainy today the probability it'll be sunny tomorrow is 0.4. Weather changes are well described by a Markov chain. (a) It's sunny today, and tomorrow is the start a 4 - day holiday. What is the probability that all 4 days are sunny? (b) A parade scheduled for the last day of the holiday. What is probability of rain on the parade? (c) A royal wed ding is planned for one year (365 days) from today. If it's an outdoor wedding what is the probability the wedding gets rained out

Solutions

Expert Solution

Given that

In a far away and long ago,there are only two weather states,rine and sun.

a)it' sunny today. So the probability that all four days are sunny = 0.8^4 = 0.4096

b) it's sunny today. Probability of rain on the last day can be in following ways = sunny*sunny*sunny*rainy + sunny*sunny*rainy*rainy + sunny*rainy*rainy*rainy + sunny*rainy*sunny*rainy= 0.8^3*0.2 + 0.8^2*0.2*0.6+ 0.8*0.2*0.6^2 + 0.8"0.2*0.4*0.2= 0.2624

If it's raining today, probability of rain on the fourth day = rain*rain*rain*rain + rain*sunny*sunny*rainy + rain*sunny*rainy*rainy + rainy*rainy*sunny*rainy = 0.6^4 + 0.6*0.4*0.8*0.2+ 0.6*0.4*0.2*0.6+ 0.6^2*0.4*0.2= 0.2256

Thus total probability of rain on the last day = 0.2624+0.2256= 0.488

c) Here since 365 days is a long period of time. We will consider the long term probability. And we find that probability of raining out is 0.33

orchestra answered 1 year ago

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