Question

Let P(A) = 0.66, P(B | A) = 0.51, and P(B | A^c) = 0.27. Use a probability tree to calculate the following probabilities: (Round your answers to 3 decimal places.)

a.	P(Ac)
b.	P(A ∩ B)
	P(Ac ∩ B)
c.	P(B)
d.	P(A | B)

Answer 1

Given that ,

P(A) = 0.66 , P(B/A) = 0.51, P(B/Ac) = 0.27

a)   P(Ac) = 1 - P(A) = 1 - 0.66 = 0.34

b) P(A ∩ B) :

P(B/A) = P(A ∩ B) / P(A)

P(A ∩ B) = P(B/A) * P(A)

P(A ∩ B) = 0.51 * 0.66 = 0.3366

P(A ∩ B) = 0.3366

P(Ac ∩ B) :

P(B | Ac) = P(Ac ∩ B) / P(Ac)

P(Ac ∩ B) =  P(B | Ac) * P(Ac)

  P(Ac ∩ B) = 0.27 * 0.34 = 0.0918

  P(Ac ∩ B) = 0.0918

c) P(B) = P(Ac ∩ B) + P(A ∩ B)

=0.0918 + 0.3366 = 0.4284

  P(B) = 0.4284

d) P(A | B) =  P(A ∩ B) / P(B)

= 0.3366 / 0.4284 = 0.7857

P(A | B) = 0.7857