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# 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 asA = (the first bulb is defective)B = (the second bulb is non – defective) C = (the two bulbs are both defective or both non defective)Determine whether(i) A, B, C are pair wise independent(ii) A, B, C are independent.

Grade:10

## 1 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
Hello Student,
Please find the answer to your question
Let S = defective and Y = non defective. Then all possible outcomes are {XX, XY, YX, YY}
Also P (XX) = 50/100 x 50/100 = 1/4,
P (XY) = 50/100 x 50/100 = 1/4,
P (YX) = 50/100 x 50/100 = 1/4,
P (YY) = 50/100 x 50/100 = 1/4
Here, A =XX ∪ XY; B = XY ∪ YY; C = XX ∪ YY
∴ P (A) = P (XX) + P (XY) = 1/4 + 1/4 = 1/2
∴ P (B) = P (XY) + P (YX) = 1/4 + 1/4 = 1/2
P (C) = P (XX) + P (YY) = 1/4 + 1/4 =1/2
Now, P (AB) = P (XY) = 1/4 = P (A). P (B)
∴ A and B are independent events.
P (BC) = P (YX) = 1/4 = P (B). P (C)
∴ B and C are independent events.
P (CA) = P (XX) = 1/4 = P (C). P (A)
∴ C and A are independent events.
P (ABC) = 0 (impossible event)
≠ P (A) P (B) P (C)
∴ A, B, C are dependent events,
Thus we can conclude that A, B, C are pair wise independent bet A, B, C are dependent events.

Thanks
Jitender Pal
askIITians Faculty

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