Question

For three events A, B and C given that
*A and C are independent.
*B and C are independent.
*A and B are disjoint.
*P(AUC)=2/3,P(BUC)=3/4,p(AUBUC)=11/12.

Find P(A), P(B), and P(C).​

Answer 1

Step-by-step explanation:

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Answer 2

P(A) = 1/3

P(B) = 1/2

P(C) = 1/2

Step-by-step explanation:

Given:

For three events A, B and C

A and C are independent.

B and C are independent.

A and B are disjoint.

P(AUC)=2/3

P(BUC)=3/4

P(AUBUC)=11/12

To find:

P(A), P(B), P(C)

Solution:

Since A and B are disjoint

Therefore, A∩B = ∅

Also A, B C will be disjoint

Therefore,

P(A∩B∩C) = 0

or, P(A∩B) = 0

A and C are independent

∴ P(A∩C) = P(A).P(B)

Similarly,

P(B∩C) = P(B).P(C)

Let P(A) = x, P(B) = y and P(C) = z

Then

P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B)-P(B∩C)-P(C∩A)+P(A∩B∩C)

11/12 = x + y + z + 0 - xy - yz + 0

or, 11/12 = x + y + z - xy - yz

or, x + y + z - xy - yz = 11/12   ............ (1)

Also

P(A∪C) = P(A) + P(C) - P(A∩C)

or, 2/3 = x + z - xz

or, x + z - xz = 2/3       ........................ (2)

And

P(B∪C) = P(B) + P(C) - P(B∩C)

or, 3/4 = y + z - yz  

or, y + z - yz = 3/4            ................. (3)

Adding (2) and (3)

x + y + 2z - xz - yz = 2/3 + 3/4

or, x + y + 2z -xz - yz = 17/12     .................. (4)

Subtracting (1) from (4)

z = 17/12 - 11/12

or, z = 6/12

or, z = 1/2

Thus, from (2)

x + 1/2 - x/2 = 2/3

or, x/2 = 2/3 - 1/2

or, x/2 = 1/6

or, x = 1/3

From (3)

y + 1/2 - y/2 = 3/4

or, y/2 = 3/4 - 1/2

or, y/2 = 1/4

or, y = 1/2

Therefore,

P(A) = 1/3

P(B) = 1/2

P(C) = 1/2

Hope this answer is helpful.

Know More:

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