Question

Suppose the weather of the day for the future days (starting from tomorrow) is good (including sunny, cloudy and etc) has probability 1/3 and the weather of the day is bad (including raining, foggy, windy and etc.) has probability 2/3.

Starting from tomorrow, compute the expectation of number of days until there are 7 consecutive days of good weather.

Answer 1

We are given here that:
P( good weather) = 1/3 and P( bad weather ) = 2/3

Let the expected number of days it takes to reach 7 consecutive days of good weather be X.

Let the expected number of days be A₁ when 1 good weather day has already occurred.
Let the expected number of days be A₂ when 2 good weather day has already occurred.
Let the expected number of days be A₃ when 3 good weather day has already occurred.
Let the expected number of days be A₄ when 4 good weather day has already occurred.
Let the expected number of days be A₅ when 5 good weather day has already occurred.
Let the expected number of days be A₆ when 6 good weather day has already occurred.

Therefore, X = (1/3)(A₁ + 1) + (2/3)*(X + 1)
3X = A₁ + 1 + 2X + 2
X = A₁ + 3

From 1 good weather already occurred, we have here:
A₁ = (2/3)*(X + 1) + (1/3)*(A₂ + 1)
3A₁ = 2X + 2 + A₂ + 1
3A₁ = 2A₁ + 6 + 2 + A₂ + 1
A₁ = A₂ + 9

From 2 good weather already occurred, we have here:
A₂ = (2/3)*(X + 1) + (1/3)*(A₃ + 1)
3A₂ = 2X + 2 + A₃ + 1
3A₁ - 27 = 2A₁ + 6 + 2 + A₃ + 1
A₁ = A₃ + 36

From 3 good weather already occurred, we have here:
A₃ = (2/3)*(X + 1) + (1/3)*(A₄ + 1)
3A₃ = 2X + 2 + A₄ + 1
3(A₁ - 36) = 2A₁ + 6 + 2 + A₄ + 1
A₁ = 108 + 9 + A₄
A₁ = A₄ + 117

and so on.. every time we are multiplying the number by 3 and adding 9 here.
A₁ = A₅ + 3*117 + 9
A₁ = A₅ + 360

A₁ = A₆ + 3*360 + 9
A₁ = A₆ + 1089

A₆ = (2/3)(X + 1) + (1/3)*1
3A₆ = 2X + 2 + 1
3A₁ - 3267 = 2A₁ + 6 + 3
A₁ = 3276

Therefore, X = 3276 + 3 = 3279

Therefore 3279 is the expected number of days until there are 7 consecutive days of good weather.