A bag contains four tickets marked with numbers 112, 121, 211, 222, One ticket is drawn at random from the bag. Let $E_{i} (i = 1, 2, 3)$ denote the event that ith digit on the ticket is 2. Then which of the following is not true?

E_{1}

and

E_{2}

are independent

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E_{2}

and

E_{3}

are independent

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E_{3}

and

E_{1}

are independent

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E_{1}, E_{2}, E_{3}

are independent

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Solution

The correct option is D $E_{1}, E_{2}, E_{3}$ are independent
$P (E_{1}) = P (E_{2}) = P (E_{3}) = \frac{1}{2}$ ,
$P (E_{i} \cap E_{j}) = 1 / 4 = P (E_{i}) P (E_{j}) \forall i \neq j$
And
$P (E_{1} \cap E_{2} \cap E_{3}) = 1 / 4 \neq P (E_{1}) P (E_{2}) P (E_{3})$

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