What is the P(A n B) and P(B/A) if P(A)=0.5, P(A u B)=0.8 and P(A/B)=0.25?

Question

What is the P(A n B) and P(B/A) if you are given these:

P(A)=0.5
P(A u B)=0.8 and
P(A/B)=0.25

Please show your work!!!!

Answer 1

P(A)=0.5, P(A U B)=0.8 and P(A/B)=0.25

P(A/B)= P(A∩B)/P(B) = 0.25

= P(A∩B)= 0.25 P(B)------------------(1)



And as P(AUB)= P(A)+ P(B) - P(A∩B)

So, 0.8= 0.5+ P(B) - P(A∩B)

P(A∩B)= P(B)-0.3---------------------(2)



From equation 1, put the value of P(A∩B) in equation 2

0.25 P(B)= P(B)-0.3

0.75 P(B)= 0.3

P(B) = 0.3/0.75 = 30/75= 0.4-------------(3)



Now put the value of P(B) in equation 2

P (A∩B)= 0.4-0.3=0.1--------------------(4)



Now, P(B/A)= P(B∩A)/P(A)

= 0.1/ 0.5

= 0.2----------------(5)



So, equation 4 and 5 are your answers

Answer 2

P(A/B) = P(A) - P(A n B)

0.25 = 0.5 - P(A n B)

P(A n B) = 0.25



P(A u B) = P(A) + P(B/A)

0.8 = 0.5 + P(B/A)

P(B/A) = 0.3

Answer 3

P(A∪B) = P(A) + P(B) - P(A∩B)

P(B) - P(A∩B) = P(A∪B) - P(A)

P(B) - P(A∩B) = 0.8 - 0.5 = 0.3



P(A∩B) = P(B) x P(A|B)

P(A∩B) = P(B) x 0.25



Solving simultaneouly the following equations, you find P(B) and P(A∩B)



P(B) - P(A∩B) = 0.3

P(A∩B) = P(B) x 0.25



P(B) - P(B) x 0.25 = 0.3

P(B)(1 - 0.25) = 0.3

P(B) x 0.75 = 0.3

P(B) = 0.3/0.75 = 0.4



P(A∩B) = 0.4 x 0.3 = 0.12



P(B|A) = P(A∩B)/P(A) = 0.12/0.5

P(B|A) = 0.24



Answers: P(A∩B) = 0.12, P(B|A) = 0.24

Answer 4

P(AuB) = P(A) + P(B) - P(AnB)

0.8 = 0.5 + P(B) - P(AnB)

0.8 = 0.5 + P(B) - P(B)*P(A/B)

0.8 = 0.5 + P(B) - P(B)*0.25

P(B) = 0.3/0.75 = 0.4

P(AnB) = P(B)*P(A/B) = 0.4*0.25 = 0.1

P(B/A) = P(AnB)/P(A) = 0.1/0.0.5 = 0.2