# Solve the following: If P(A ∩ B) = 12, P(B ∩ C) = 13, P(C ∩ A) = 16 then find P(A), P(B) and P(C) - Mathematics and Statistics

Sum

Solve the following:

If P(A ∩ B) = 1/2, P(B ∩ C) = 1/3, P(C ∩ A) = 1/6 then find P(A), P(B) and P(C)

#### Solution

Since A and B are independent events,

P(A ∩ B) = P(A) . P(B)

∴ P(A) P(B) = 1/2   ...(i)

B and C are independent events.

∴ P(B ∩ C) = P(B) . P(C)

∴ P(B) P(C) = 1/3   ...(ii)

A and C are independent events.

∴ P(A ∩ C) = P(A) . P(C)

∴ P(A) P(C) = 1/6  ...(iii)

Dividing (i) by (ii), we get

("P"("A")"P"("B"))/("P"("B")"P"("C")) = (1/2)/(1/3)

∴ P(A) = 3/2 P(C)   ...(iv)

Substituting equation (iv) in (iii), we get

3/2P(C) . P(C) = 1/6

∴ [P(C)]2 = 1/9

∴ P(C) = 1/3

Substituting P(C) = 1/3 in equation (ii), we get P(B) = 1

Substituting P(B) = 1 in equation (i), we get P(A) = 1/2

Concept: Independent Events
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 9 Probability
Miscellaneous Exercise 9 | Q II. (4) | Page 213