Weather Forecast

Probability Level 3

In Malaysia, we don't have four seasons. Everyday, it's either sunny, windy or rainy. After living in this lovely country for 19 years, I've got to a conclusion :

  • We never have two rainy days in a row. (Hooray!)
  • If we have a rainy day, it's just as likely to have a sunny day as a windy day tomorrow.
  • If we have a sunny or a windy day, there's an even chance of having the same the next day.
  • If there is change from sunny or windy, only half of the time is this a change to a rainy day.

Given that today is a rainy day, then which of the following is more likely to occur?

It's more likely to have a sunny day after a week It's more likely to have a windy day after a week They are both equally likely

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Christopher Boo by Christopher Boo , 24, Singapore

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2 solutions

Just my intuition:

Notice that all the conditions given are "symmetrical". So it is "intuitive" that after week, both would be equally likely.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Christopher Boo
Jun 16, 2016

Relevant wiki: Markov Chain

Consider the transition matrix of the weather,

T = ( 0 1 2 1 2 1 4 1 2 1 4 1 4 1 4 1 2 ) T = \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{2} \\ \end{pmatrix}

  • The first row describes the probability to have a rainy day, sunny day and windy day after a rainy day.
  • The second row describes the probability to have a rainy day, sunny day and windy day after a sunny day.
  • The third row describes the probability to have a rainy day, sunny day and windy day after a windy day.

The probability of having a windy day two days later after a sunny day is

T 21 × T 13 + T 22 × T 23 + T 23 × T 33 T_{21}\times T_{13} + T_{22}\times T_{23} + T_{23}\times T_{33}

In general, the probability is the entry of T 23 2 T^2_{23} .

With this observation, T 12 7 T^7_{12} and T 13 7 T^7_{13} is the probability that it will have a sunny and windy day after a week (7 days) from a rainy day.

T 7 = ( 1 5 2 5 2 5 1 5 2 5 2 5 1 5 2 5 2 5 ) T^7 = \begin{pmatrix} \frac{1}{5} & \frac{2}{5} & \frac{2}{5} \\ \frac{1}{5} & \frac{2}{5} & \frac{2}{5} \\ \frac{1}{5} & \frac{2}{5} & \frac{2}{5} \\ \end{pmatrix}

They are the same!

1 Helpful 0 Interesting 0 Brilliant 0 Confused

For more information, this is a good lecture note.

Christopher Boo - 4 years, 12 months ago

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