Problem 121 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A bag contains four tickets with numbers 112,121 , \(211,222 .\) One ticket is drawn at random from the bag. Let \(\mathrm{E}_{j}(i=1,2,3)\) denote the event that \(i\) th digit on the drawn ticket is \(2 .\) Then, (A) \(E_{1}, E_{2}, E_{3}\) are pair-wise independent (B) \(E_{1}, \bar{E}_{2}\) are independent (C) \(\bar{E}_{2}\) and \(\bar{E}_{3}\) are not independent (D) \(E_{1}, E_{2}, E_{3}\) are mutually independent

Step-by-Step Solution

Answer

(A) True, (B) False, (C) True, (D) False.

Step 1: Define the Events

Let's define each event. \(E_1\) is the event that the 1st digit is 2, \(E_2\) the event that the 2nd digit is 2, and \(E_3\) the event that the 3rd digit is 2. The complement \(\bar{E}_2\) is the event that the 2nd digit is not 2, and \(\bar{E}_3\) is the event that the 3rd digit is not 2.

Step 2: List All Possible Outcomes

The tickets are numbered as follows: 112, 121, 211, and 222. Therefore, our sample space \(S\) contains these four possibilities.

Step 3: Calculate Probability of Each Event

Find \(P(E_1)\), \(P(E_2)\), and \(P(E_3)\). For each event, count how many tickets satisfy the condition and divide by the total number of tickets. For example, \(P(E_1) = \frac{2}{4} = \frac{1}{2}\) because there are two tickets (211, 222) where the 1st digit is 2.

Step 4: Check Pair-Wise Independence

Events \(E_1\), \(E_2\), and \(E_3\) are pair-wise independent if \(P(E_i \cap E_j) = P(E_i)\cdot P(E_j)\) for all pairs \(i eq j\). Calculate these for every pair of \(E_1\), \(E_2\), and \(E_3\).

Step 5: Check Independence of Complements

To determine if \(E_1\) and \(\bar{E_2}\) are independent, find \(P(E_1 \cap \bar{E_2})\) and check whether it equals \(P(E_1) \cdot P(\bar{E_2})\). Also calculate \(P(\bar{E_2} \cap \bar{E_3})\) and verify if it equals \(P(\bar{E_2}) \cdot P(\bar{E_3})\).

Step 6: Check Mutual Independence

The events \(E_1\), \(E_2\), and \(E_3\) are mutually independent if any combination of these follows \(P(E_1 \cap E_2 \cap E_3) = P(E_1) \cdot P(E_2) \cdot P(E_3)\). Verify this condition to check mutual independence.

Step 7: Determine Which Statements Are True

After calculating and verifying each scenario, compare the results against the provided options (A), (B), (C), and (D) to determine which are true.

Key Concepts

Pair-wise independence

Understanding pair-wise independence in probability involves focusing on just two events at a time. If we're looking at events like drawing a 2 in specific positions on a ticket, pair-wise independence checks whether the occurrence of one event doesn't affect the occurrence of another. For example, let's take two events, say, the first digit and the second digit being a 2. Pair-wise independence requires:

• Calculate the probability of each individual event, such as \(P(E_1)\) and \(P(E_2)\).
• Find the probability that both events occur together, \(P(E_1 \cap E_2)\).
• If \(P(E_1 \cap E_2) = P(E_1) \cdot P(E_2)\), the events are pair-wise independent.

This approach checks for independence across every possible pair with our given events.

Mutual independence

Mutual independence is a more stringent requirement than pair-wise. It involves all events occurring together without any one event impacting the others. This means each event's occurrence does not alter the likelihood of the combination of all other events in the set. To achieve mutual independence among events like \(E_1\), \(E_2\), and \(E_3\), you'll need to prove that:

• The probability of all events happening simultaneously \(P(E_1 \cap E_2 \cap E_3)\) is equal to the product of their individual probabilities, \(P(E_1) \cdot P(E_2) \cdot P(E_3)\).

This condition ensures total independence, not just among pairs but among all combinations of events.

Probability calculation

Probability calculation is fundamental in verifying both pair-wise and mutual independence. It starts with determining how likely specific outcomes are from a defined sample space. Follow these steps for calculating probabilities:

• Identify the total number of outcomes in the sample space. For the given tickets, there are four outcomes.
• For each event, count favorable outcomes — like how many tickets meet the event criteria.
• Divide the number of favorable outcomes by the total number of outcomes to calculate the probability. For instance, if two tickets have the first digit as 2, \(P(E_1) = \frac{2}{4} = \frac{1}{2}\).

Calculate the intersection of events for compound probabilities considering both pair-wise and mutual scenarios. This ensures a complete understanding of how events interact probabilistically.