statistics 2 The prevailing wind direction at a building sit

2. The prevailing wind direction at a building site takes values from the set D = {0 degree,30 degree,60 degree,90 degree} (0 degree indicates a wind from due north, 90 degree indicates a wind from due east), while the wind speed takes values from the set V = {0 kph ,25 kph ,50 kph,75 kph,100 kph}. (a) Represent the events: E1={(D,V):V > 35kph} E2 {(D,V): l5kph

Solution

(a)

E₁ = { (D,V) : V>35 kph}

     = {(0⁰,50kph), (0⁰,75kph), (0⁰,100kph),

          (30⁰,50kph), (30⁰,75kph), (30⁰,100kph),

          (60⁰,50kph), (60⁰,75kph), (60⁰,100kph),

          (90⁰,50kph), (90⁰,75kph), (90⁰,100kph) }

E₂ = { (D,V) : 15kph < V < 80kph}

     = { (0⁰,25kph), (0⁰,50kph), (0⁰,75kph),

          (30⁰,25kph), (30⁰,50kph), (30⁰,75kph),

          (60⁰,25kph), (60⁰,50kph), (60⁰,75kph),

          (90⁰,25kph), (90⁰,50kph), (90⁰,75kph) }

E₃ = { (D,V) : D30⁰ }

     = { (0⁰,0 kph), (30⁰,0 kph),

          (0⁰,25 kph), (30⁰,25 kph),

          (0⁰,50 kph), (30⁰,50 kph),

          (0⁰,75 kph), (30⁰,75 kph),

          (0⁰,100 kph), (30⁰,100 kph) }

E₄ = { (D,V) : Vsin(D) >60 }

     = { (60⁰,75kph), (60⁰,100 kph),

          (90⁰,75kph), (90⁰,100 kph) }

A = E₁ E₂

    = { (D,V) : V>35 kph   (D,V) : 15kph < V < 80kph}

    = {(D,V) : 35kph < V < 80kph }

    = { (0⁰,50 kph), (0⁰,75 kph),

          (30⁰,50 kph), (30⁰,75 kph),

          (60⁰,50 kph), (60⁰,75 kph),

          (90⁰,50 kph), (90⁰,75 kph) }

B = E₁ U E₃

    = { (D,V) : V>35 kph   U (D,V) : D30⁰}

    = {(0⁰,50kph), (0⁰,75kph), (0⁰,100kph),

          (30⁰,50kph), (30⁰,75kph), (30⁰,100kph),

          (60⁰,50kph), (60⁰,75kph), (60⁰,100kph),

          (90⁰,50kph), (90⁰,75kph), (90⁰,100kph),

          (0⁰,0 kph), (30⁰,0 kph),(0⁰,25 kph),

          (30⁰,25 kph)}

C = A B

   = { (0⁰,50 kph), (0⁰,75 kph),

          (30⁰,50 kph), (30⁰,75 kph),

          (60⁰,50 kph), (60⁰,75 kph),

          (90⁰,50 kph), (90⁰,75 kph) }

(b)

Here total no. of possible combinations of wind direction and speed is N= 4*5 = 20

P(A) = n(A)/N

          = 8/20

          = .4

P(B) = n(B)/N

          = 16/20

          = .8

P(C) = n(C)/N

          = 8/20

          = .4

P(A U B) = P(A) + P(B) – P(AB)

                = P(A) + p(B) – P(C)

               = .4 + .8 -.4

              = .8

P(E₄) = n(E₄)/N

          = 2/20

          = .1