Mathematics?

Question

If P(A)=1/2, P(A'/B)=1/3 and P(AUB)=3/5, determine the values of P(AnB), P(B/A') and P(A/B').

Answer 1

Conditional probability :

If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by
P(A n B) = P(A)P(B/A).

Probability complement : The probability of complement of event A : P(A ') = 1 - P(A).

Given : P(A) = 1/2,

              P(A '/B) = 1/3, and

              P(A U B) = 3/5.

• P(A n B).

Use the formula : P(A U B) = P(A) + P(B) - P(A n B).

⇒ P(A n B) = P(A) + P(B) - P(A U B).

P(A '/B) = P(A ' n B)/P(B) = 1/3

P(A ' n B)/P(B) = 1/3

⇒ P(A ' n B) = P(B)/3.

Draw Venn diagram with 2 sets, i. e,  A , B and fill in the regions in terms of P(B)
---> (1/2)+P(B)/3 = 3/5 --->P(B)=3/10

⇒ P(A n B) = P(A) + P(B) - P(A U B)

= 1/2 + 3/10 - 3/5

= 1/5.

∴  P(A n B) = 1/5.

• P(B/A ') = P(A ' n B)/P(A ').

P(A ∩ B) + P(A' ∩ B) = P(B)

(1/5) + P(A' ∩ B) = 3/10

P(A' ∩ B) = 3/10 - 1/5

P(A' ∩ B) = (3 - 2)/10 = 1/10

⇒ P(B/A ') = P(A ' n B)/P(A ')

P(A) = 1/2 ⇒P(A ') = 1 - 1/2 = 1/2.

= (1/10)/(1/2)

= 1/5.

∴  P(B/A ') = 1/5.

• P(A/B ') = P(A n B ')/P(B ').

P( A ∩ B') + P(A ∩ B) = P(A)

P( A ∩ B') + (1/5) = 1/2

P( A ∩ B') = 1/2 - 1/5

P( A ∩ B') = 3/10.

P(B) = 3/10 ⇒P(B ') = 1 - 3/10 = 7/10

⇒ P(A/B ') = P(A n B ')/P(B ')

= (3/10)/(7/10)

= 3/7.

∴  P(A/B ') = 3/7.