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Complement of an Event

A bag contain...

Question

A bag contains four tickets marked 112, 121, 211, 222. One ticket is drawn at random from the bag.
Let $E_{i} (i = 1, 2, 3)$ denote the event that $i^{t h}$ digit on the ticket is 2. Which of the following option(s) is/are correct?

E_{1}

and

E_{2}

are independent

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E_{2}

and

E_{3}

are independent

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E_{1}

and

E_{3}

are independent

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E_{1}, E_{2}

and

E_{3}

are independent

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Solution

The correct options are
A $E_{1}$ and $E_{2}$ are independent
B $E_{2}$ and $E_{3}$ are independent
C $E_{1}$ and $E_{3}$ are independent
We have $P (E_{i}) = \frac{2}{4} = \frac{1}{2} for i = 1, 2, 3 Also for i \neq j, P (E_{i} \cap E_{j}) = \frac{1}{4} = P (E_{i}) P (E_{j}) Therefore, E_{i} and E_{j} are independent for i \neq j Also, P (E_{1} \cap E_{2} \cap E_{3}) = \frac{1}{4} \neq P (E_{1}) P (E_{2}) P (E_{3}) ∴ E_{1}, E_{2}, E_{3} are not independent events.$

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A bag contains four tickets marked 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. Which of the following option(s) is/are correct?

A bag contains four tickets marked 112, 121, 211, 222. One ticket is drawn at random from the bag.
Let $E_{i} (i = 1, 2, 3)$ denote the event that $i^{t h}$ digit on the ticket is 2. Which of the following option(s) is/are correct?