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A bag contains four tickets having numbers 112, 121, 211, 222 written on them. Denote by $A_{i} (i = 1, 2, 3)$ the event that the $i^{t h}$ digit from left of the number on a randomly drawn ticket is 1.
Which of the following is true?

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$A_{1}, A_{2}, A_{3}$ are independent events

$P (A_{i} A_{j}) = P (A_{i}) P (A_{j}) f o r a l l i \leq j$

$P (A_{i} A_{j}) \neq P (A_{i}) P (A_{j}) f o r a l l 1 \leq i < j \leq 3$

$P (A_{i} A_{j}) = P (A_{i}) P (A_{j}) f o r a l l 1 \leq i < j \leq 3$

answer is D.

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Detailed Solution

$P (A_{1}) = \frac{1}{2}$ , $P (A_{2}) = \frac{1}{2}$ and $P (A_{3}) = \frac{1}{2}$

$P (A_{1} \cap A_{2}) = P (A_{1}) \cdot P (A_{2})$ and so on

Events are pair wise independent events

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