Probability
Probability
Allen Star Batch
Grade 12
Question:
A bag contains four tickets marked with numbers 112, 121, 211, and 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:
$E_1$ and $E_2$ are independent
$E_2$ and $E_3$ are independent
$E_3$ and $E_1$ are independent
$E_1, E_2, E_3$ are independent
Step-by-Step Solution
Key Concept: Events are independent when $P(E_i \cap E_j) = P(E_i)P(E_j)$ for all pairs and $P(E_1 \cap E_2 \cap E_3) = P(E_1)P(E_2)P(E_3)$ for the triple product.
Let $E_1$ be the event that the first digit is 2 (numbers 211 or 222): $P(E_1) = \frac{2}{4} = \frac{1}{2}$. Let $E_2$ be the event that the second digit is 2 (numbers 121, 222): $P(E_2) = \frac{1}{2}$. Let $E_3$ be the event that the third digit is 2 (numbers 222, 112): $P(E_3) = \frac{1}{2}$. Since $E_1 \cap E_2$ gives 222 with $P(E_1 \cap E_2) = \frac{1}{4} = P(E_1)P(E_2)$, and similarly for other pairs and the triple intersection, the events are independent.
Correct Answer: 1,2,3