Question

For a given day, temperature can be recorded as hot, mild and cold while weather can be sunny or cloudy. The probability that the temperature is hot, mild and cold are .15, .55 and .30, respectively. Probability that the weather is sunny if the temperature is hot is .67, if the temperature is recorded mild is.36, and if the temperature is cold is.33. Show all the steps including identification of what formulas/properties you used. Points will be deducted from answers if only the final answer is provided.
(a) List all probabilities given in this problem, expressed in terms of events that you define.
(b) What is the probability that it is a sunny day?
(c) Given that the weather is sunny, what is the probability that the temperature is hot? mild? cold?
(d) Given that the weather is cloudy, what is the probability that the temperature is hot? mild? cold?
(e) Is the cloudy weather independent of hot weather?

Answer 1

Answer:

a) p(H) = 0.15

,   p(M) = 0.55

,   p(C) = 0.30,

p(S | H) i.e. sunny and hot  = 0.67

P(c | H) i.e. cloudy and hot  = 1 - 0.67 = 0.33

P( S | M) i.e. sunny and mild = 0.36

P(c | M) i.e. cloudy and mild  = 1 - 0.36 = 0.64

P( S | C )i.e. sunny and cold  = 0.33

P( c | C)   = 1 - 0.33 = 0.67

b) ≈ 0.40

c) P ( H | S )  = 0.2513

  P( M | S )   = 0.495

  P(C | S) = 0.2475

d) P( H I c ) = 0.0825

Probability of mild and cloudy= 0.587

Probability of cold and cloudy  = 0.335

e)  No

Step-by-step explanation:

a) List All probabilities given in this problem

lets represent each condition with the first letter

Temperature : Hot ( H ) , mild ( M ) , cold ( C )

weather : Sunny ( S ) and cloudy ( c )

hence the probabilities are ;

p(H) = 0.15,   p(M) = 0.55,   p(C) = 0.30,

p(S | H) i.e. sunny and hot  = 0.67

P(c | H) i.e. cloudy and hot  = 1 - 0.67 = 0.33

P( S | M) i.e. sunny and mild = 0.36

P(c | M) i.e. cloudy and mild  = 1 - 0.36 = 0.64

P( S | C )i.e. sunny and cold  = 0.33

P( c | C) i.e. cloudy and cold  = 1 - 0.33 = 0.67

b) probability of a sunny day

P ( s ) =  P( H ) * P( S | H ) +  P( M ) * P( S | M ) + P(C) * P( S | C)

          = (0.15 * 0.67) + ( 0.55 * 0.36 ) + ( 0.30 * 0.33 )

           = 0.3975 ≈ 0.40

c) Given the weather is sunny ( application of Bayes theorem = P(A and B) / P(B) )

P ( H | S ) = ( 0.15* 0.67 ) / 0.4  

               = 0.2513

P( M | S ) = 0.55 * 0.36 / 0.3975  

               = 0.198 / 0.4

               = 0.495

P(C | S) = 0.3 * 0.33 / 0.4

            = 0.099  = 0.2475

D) Given that the weather is cloudy

P( c ) = 1 - 0.4    = 0.6

P( H I c ) = ( 0.15*0.33) / 0.6

= 0.0825

Probability of mild and cloudy = (0.55*0.64) / 0.6  = 0.587

Probability of cold and cloudy  = ( 0.30 * 0.67) / 0.6  = 0.335

e) The cloudy weather ( c ) is not Independent of the hot weather

because condition for independency ( P(c) x P( H )  = P( c and H) ) is not fulfilled.