A bag contains four tickets marked with 112,121,211,222, one ticket is drawn at random from the bag. Let Ei(i = 1, 2, 3) denote the event that ith digit on the ticket is 2 then :

Engineering

Mathematics

Conditional Probability

Question

A bag contains four tickets marked with 112,121,211,222, one ticket is drawn at random from the bag. Let E_i(i = 1, 2, 3) denote the event that i^th digit on the ticket is 2 then :

E₁ and E₂ are independent

E₂ and E₃ are independent

E₃ and E₁ are independent

E₁, E₂, E₃ are independent

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$P (E_{1}) = P (E_{2}) = P (E_{3}) = \frac{1}{2}$

$P (E_{i} \cap E_{j}) = \frac{1}{4} = P (E_{i}) P (E_{j})$
Hence, two events taken together are independent.
$P (E_{1} \cap E_{2} \cap E_{3}) = \frac{1}{4} \neq P (E_{1}) P (E_{2}) P (E_{3})$
Therefore, the three events are not independent together.
Hence, options A, B and C are correct.

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