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Let two fair six-faced dice A and B be thrown simultaneously. If `E_1` is the event that die A shows up four, `E_2` is the event that die B shows up t

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Let two fair six-faced dice A and B be thrown simultaneously. If E1 is the event that die A shows up four, E2 is the event that die B shows up two and E3 is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true ? (1) E1 and E2 are independent. (2) E2 and E3 are independent. (3) E1 and E3 are independent. (4) E1 , E2 and E3 are independent. A. E1 and E2 are independent B. E2 and E3 are independent C. E1 and E3 are independent D. E1,E2 and E3 are independent Select the correct answer from above options

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Correct Answer - D We find that P(E1)= 6 36 ,P(E2)= 6 36 = 1 6 , P(E3)= 1 2 ,P(E1∩E2)= 1 36 ,P(E2∩E3)= 3 36 = 1 12 , P(E1∩E3)= 3 36 = 1 12 and P(E1∩E2∩E3)=0 Clearly, P(E1∩E2)=P(E1)P(E2)⇒E1,E2 are independent P(E1∩E3)=P(E1)P(E3)⇒E1,E3 are independent But, P(E1∩E2∩E3)≠P(E1)P(E2)P(E3) ⇒E1,E2,E3 are not independent Thus, option (d) is incorrect.

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