Engineering
Mathematics
Conditional Probability
Question

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.

Only (1) is true.

Both (1) and (2) are true.

Only (2) is true.

Both (1) and (2) are false.

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Solution
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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)}=50100×1=12

P(B)={P(D)+P(D)}×P(D)=1×50100=12P(C)=P(DDDD)=P(DD)+P(DD)P(DDDD)
P(C)=12×12+12×120=12
Now
P(AB)=P(DD)=50100×50100=14
P(BC)=P(DD)=12×12=14
P(CA)=P(DD)=12×12=14
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)
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