A lot contains 50 defective and 50 non-defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events A, B, C are defined as A = {the first bulb is defective}, B = {the second bulb is non defective}, C = {the two bulbs are both defective or both non defective}, then which of the following statements is/are true ? (1) A, B, C are pair wise independent. (2) A, B, C are independent.

Engineering

Mathematics

Basic Definitions Related to Probability

Question

Both (1) and (2) are false.

Only (1) is true.

Only (2) is true.

Both (1) and (2) are true.

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Let's denote the defective item by D and non-defective by $D$

$P (A) = P (D) {P (D) + P (D)} = \frac{50}{100} \times 1 = \frac{1}{2}$

P (B) = {P (D) + P (D)} \times P (D) = 1 \times \frac{50}{100} = \frac{1}{2}

P (C) = P (DD \cup D D) = P (DD) + P (D D) - P (DD \cap D D)

\Rightarrow P (C) = \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} - 0 = \frac{1}{2}

Now

P (A \cap B) = P (D D) = \frac{50}{100} \times \frac{50}{100} = \frac{1}{4}

P (B \cap C) = P (D D) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

P (C \cap A) = P (DD) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

Thus, we can see that

P(A∩B) = P(A)P(B), P(B∩C) = P(B)P(C)

P(C∩A) = P(C)P(A)