5 Use probability tree Given A GM stock B Dow Jones Index

5. Use probability tree: Given A = GM stock ?, B = Dow Jones Index ?, the p(A) = .8 and both the index and GM stock can only go up or down. Also, p(B / A) = .7 and the p(B / A^C) = .25.

Find each of the following:

a) p(A^C) =

b) p(B^C / A) =

c) p(B^C / A^C) =

d) p(AB) =

e) p(A^CB) =

f) p(B) =

g) p(B^C) =

h) p(A u B) =

i) p(A^C u B^C) =

j) p(A / B) =

k) p(A^C / B^C) =

l) Are A and B independent?  

Solution

a) p(A^C) = 1-P(A)= 1-0.8 =0.2

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b) p(B / A) = .7

--> P(A and B)/P(A) = 0.7

So P(A and B)= 0.7*0.8= 0.56

p(B^C / A) = P(B\' and A)/P(A)

={P(A)-P(A and B)}/ P(A)

=(0.8-0.56)/0.8

=0.3

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c) p(B / A^C) = .25

--> P(B and A\')/P(A\')= 0.25

So P(B and A\')= 0.25*0.2= 0.05

--> P(B) - P(A and B) =0.05

So P(B) = 0.05+0.56=0.61

p(B^C / A^C) = P(B\' and A\')/P(A\')

=(1-P(A or B))/P(A\')

=(1-P(A)-P(B)+P(A and B)) / P(A\')

=(1-0.8-0.61+0.56)/0.2

=0.75

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d) p(AB) = 0.56

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e) p(A^CB) =P(B)- P(A and B) = 0.61-0.56 =0.05

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f) p(B) =0.61

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g) p(B^C) = 1-P(B)=1-0.61= 0.39

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h) p(A u B) = P(A)+ P(B)-P(A and B)

=0.8+0.61-0.56

=0.85

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i) p(A^C u B^C) = 1-P(A and B) =1-0.56 =0.44

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j) p(A / B) = P(A and B)/P(B)

=0.56/0.61

=0.9180328

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k) p(A^C / B^C) = P(A\' and B\')/P(B\')

=(1-P(A)-P(B)+P(A and B))/ P(B\')

=(1-0.8-0.61+0.56)/0.39

=0.3846154

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l) Are A and B independent?  

No, because P(A)*P(B)=0.8*0.61 is not equal to P(A and B)=0.56