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Sagot :
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.
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