Question

Determine all joint probabilities listed below from the following information: P(A)=0.77,P(Ac)=0.23,P(B|A)=0.37,P(B|Ac)=0.64 P(A and B) = P(A and Bc) = P(Ac and B) = P(Ac and Bc) =

Answer 1

Given that, P(A) = 0.77, P(A^{​​​​​​c}) = 0.23, P(B | A) = 0.37

P(B | A^{​​​​​​c}) = 0.64

i) P(B | A) = P(A and B) / P(A)

=> P(A and B) = P(B | A) * P(A)

=> P(A and B) = 0.37 * 0.77

=> P(A and B) = 0.2849

ii) P(A and B^{​​​​​​c}) = P(A) - P(A and B) = 0.77 - 0.2849 = 0.4851

=> P(A and B^{​​​​​​c}) = 0.4851

iii) P(B | A^{​​​​​​c}) = P(A^{​​​​​​c} and B) / P(A^{​​​​​​c})

=> P(A^{​​​​​​c} and B) = P(B | A^{​​​​​​c}) * P(A^{​​​​​​c})

=> P(A^{​​​​​​c} and B) = 0.64 * 0.23

=> P(A^{​​​​​​c} and B) = 0.1472

iv) P(A^{​​​​​​c} and B) = P(B) - P(A and B)

=> P(B) = (A^{​​​​​​c} and B) + P(A and B)

=> P(B) = 0.1472 + 0.2849

=> P(B) = 0.4321

P(A or B) = P(A) + P(B) - P(A and B)

=> P(A or B) = 0.77 + 0.4321 - 0.2849

=> P(A or B) = 0.9172

Therefore,

P(A^{​​​​​​c} and B^{​​​​​​c}) = 1 - P(A or B) = 1 - 0.9172 = 0.0828

=> P(A^{​​​​​​c} and B^{​​​​​​c}) = 0.0828