Question

In the figure below, Ω is the set of all objects. Black is the set of all black objects, and White is the set of all white objects. Square is the set of all square objects, and A is the set of all objects containing A. Use the relative frequency of occurrence and the POI (principle of indifference) to answer the following questions:

Calculate P(A) and P(A|Square). State whether or not A and Square are independent. State the reason for your answer. b) Calculate P(A|Black) and P(A|Square⋂Black). State whether or not A and Square are conditionally independent given Black. State the reason for your answer. c) Calculate P(A|White) and P(A|Square⋂White). State whether or not A and Square are conditionally independent given White. State the reason for your answer.

In the figure below, is the set of all objects. Black is the set of all black objects, and White is the set of all white objects. Square is the set of all square objects, and A is the set of all objects containing A. Use the relative frequency of occurrence and the Pol (principle of indifference) to answer the following questions: a) Calculate P(A) and P(AlSquare). State whether or not A and Square are independent. State the reason for your answer. b) Calculate P(AlBlack) and P(AlSquarenBlack). State whether or not A and Square are conditionally independent given Black. State the reason for your answer. c) Calculate P(AlWhite) and P(AlSquarenWhite). State whether or not A and Square are conditionally independent given White. State the reason for your answer.

Answer 1

total objects = 13

total square=8

circle=5

a)

P(A) = 4/13

P(A| square) = P(A and square)/P(square) = 3/8

A and square are independent if

P(A and Square)=P(A)*P(square)

P(A and Square) = 3/13

P(A)*P(Square) = 4/13*8/13

since,

P(A and Square)╪P(A)*P(square)

so, these are not independent

b)

P(A|Black) = P(A and Black)/P(Black) = 3/9 = 1/3

P(A|square n Black)=P(A n Square n Black)/P(square n black) = 2/6 = 1/3

A and square are conditionaly independent given black if

P(A|black and Square|black)=P(A|black)*P(square|black)

P(A|black) = 3/9 = 1/3

P(square|black)=4/9

P(A|black and Square|black) =P(A and square | black) = 2/9

since, P(A|black and Square|black)╪P(A|black)*P(square|black)

so, these are not independent

c)

P(A|White) = P(A and white)/P(white) = 2/4 = 1/2

P(A|square n white) = 1/2

A and square are conditionaly independent given white if

P(A and Square|white)=P(A|white)*P(square| white)

P(A|White) = P(A and white)/P(white) = 2/4 = 1/2

P(square|white) = 2/4 = 1/2

P(A and square | white) = P(A and square and white)/P(white) = 1/4

since,

P(A and Square|white)=P(A|white)*P(square| white)

so, events are independent.