Let A and B be two independent events such that P(A) = p₁ and P(B) = p₂. Describe in words the events whose probabilities are:

(i) p₁p₂ (ii) (1 $-$ p₁)p₂ (iii) 1 $-$ (1 $-$ p₁)(1 $-$ p₂) (iv) p₁ + p₂ $-$ 2p₁p₂ [NCERT EXEMPLAR]

Open in App

Solution

$(i) As, p_{1} p_{2} = P (A) \times P (B) And, A and B are independent events . i . e . P (A) \times P (B) = P (A \cap B) So, P (A \cap B) = p_{1} p_{2} Hence, p_{1} p_{2} = P (A and B occur) (ii) As, (1 - p_{1}) p_{2} = [1 - P (A)] \times P (B) = P (A) \times P (B) And, A and B are independent events . i . e . P (A) \times P (B) = P (A \cap B) So, P (A \cap B) = (1 - p_{1}) p_{2} Hence, (1 - p_{1}) p_{2} = P (A does not occur, but B occurs) (iii) As, 1 - (1 - p_{1}) (1 - p_{2}) = 1 - [1 - P (A)] \times [1 - P (B)] = 1 - P (A) \times P (B) And, A and B are independent events . i . e . P (A) \times P (B) = P (A \cap B) \Rightarrow 1 - (1 - p_{1}) (1 - p_{2}) = 1 - P (A \cap B) = 1 - P (A \cup B) = P (A \cup B) So, P (A \cup B) = 1 - (1 - p_{1}) (1 - p_{2}) Hence, 1 - (1 - p_{1}) (1 - p_{2}) = P (At least one of A and B occurs) (iv) As, p_{1} + p_{2} - 2 p_{1} p_{2} = (p_{1} - p_{1} p_{2}) + (p_{2} - p_{1} p_{2}) = [P (A) - P (A) \times P (B)] + [P (B) - P (A) \times P (B)] And, A and B are independent events . i . e . P (A) \times P (B) = P (A \cap B) \Rightarrow p_{1} + p_{2} - 2 p_{1} p_{2} = [P (A) - P (A \cap B)] + [P (B) - P (A \cap B)] = P (only A) + P (only B) So, P (only A) + P (only B) = p_{1} + p_{2} - 2 p_{1} p_{2} Hence, p_{1} + p_{2} - 2 p_{1} p_{2} = P (Exactly one of A and B occurs)$

Suggest Corrections

Similar questions

Let $E_{1}$ and $E_{2}$ be two independent events such that $P (E_{1}) = P_{1}$ and $P (E_{2}) = P_{2}$ . Describe in words of the events whose probabilities are

(i) $P_{1} P_{2}$

(ii) $(1 - P_{1}) P_{2}$

(iii) $1 - (1 - P_{1}) (1 - P_{2})$

(iv) $P_{1} + P_{2} - 2 P_{1} P_{2}$

Let $E_{1}$ and $E_{2}$ be two independent events such that $P (E_{1}) = P_{1}$ and $P (E_{2}) = P_{2}$ . Describe in words of the events whose probabilities are