Problem 6 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

We choose a month of the year, in such a manner that each month has the same probability. Find out whether the following events are independent: a. the events "outcome is an even numbered month" (i.e., February, April, June, etc.) and "outcome is in the first half of the year." b. the events "outcome is an even numbered month" (i.e., February, April, June, etc.) and "outcome is a summer month" (i.e., June, July, August).

Step-by-Step Solution

Answer

Event A and B1 are independent. Event A and B2 are not independent.

Step 1: Understand the Sample Space

The sample space, S, consists of the 12 months of the year: January, February, March, April, May, June, July, August, September, October, November, and December. Since each month is equally likely, the probability of choosing any specific month is \( \frac{1}{12} \).

Step 2: Define Event A: Even Numbered Month

Event A is defined as the months that are even-numbered: February (2), April (4), June (6), August (8), October (10), and December (12). There are 6 months in event A, so the probability \( P(A) = \frac{6}{12} = \frac{1}{2} \).

Step 3: Define Event B1: First Half of the Year

Event B1 consists of the months in the first half of the year: January, February, March, April, May, and June. There are 6 months in event B1, resulting in a probability \( P(B1) = \frac{6}{12} = \frac{1}{2} \).

Step 4: Find Intersection of Event A and Event B1

The intersection \( A \cap B1 \) consists of the even-numbered months in the first half: February, April, and June. Thus, \( P(A \cap B1) = \frac{3}{12} = \frac{1}{4} \).

Step 5: Check Independence of Event A and Event B1

Events A and B1 are independent if \( P(A \cap B1) = P(A) \times P(B1) \). Calculate \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). Since \( P(A \cap B1) = \frac{1}{4} \), events A and B1 are independent.

Step 6: Define Event B2: Summer Month

Event B2 consists of the summer months: June, July, and August. There are 3 months in event B2, giving a probability \( P(B2) = \frac{3}{12} = \frac{1}{4} \).

Step 7: Find Intersection of Event A and Event B2

The intersection \( A \cap B2 \) only includes the month June, which is both an even-numbered month and a summer month. Therefore, \( P(A \cap B2) = \frac{1}{12} \).

Step 8: Check Independence of Event A and Event B2

Events A and B2 are independent if \( P(A \cap B2) = P(A) \times P(B2) \). Calculate \( \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \). Since \( P(A \cap B2) = \frac{1}{12} \) and this is not equal to \( \frac{1}{8} \), events A and B2 are not independent.

Key Concepts

Sample Space

When we talk about probability, understanding the sample space is super important. The sample space is simply a list of all possible outcomes of an experiment. In the context of our exercise, the sample space consists of all 12 months of the year: January through December. This is because each month is a possible outcome when randomly choosing a month.

Since the problem states that each month is chosen with the same probability, we have what's called a uniform distribution. This means that the chance of picking any one specific month is equal. Mathematically, for our scenario with 12 possible outcomes, this probability is calculated as:

• Probability of any single month = \( \frac{1}{12} \)

Understanding the sample space sets the foundation for all further calculations, as it informs us of the number of outcomes and the nature of their likelihood.

Event Intersection

An intersection in probability refers to the event where two or more conditions are all satisfied simultaneously. In our exercise, we encountered intersections in examining whether two events are independent.

Let's break it down using Event A (even-numbered months) and Event B1 (months in the first half of the year). The intersection of these two events, denoted as \( A \cap B1 \), consists of the months that are both even-numbered and in the first half. This includes February, April, and June.

To find the probability of the intersection, you look at the number of favorable outcomes in both events over the total number of outcomes. In this case, there are three such months, so:

• \( P(A \cap B1) = \frac{3}{12} = \frac{1}{4} \)

The concept of event intersection is critical to understanding how different conditions interact with each other, helping us assess probabilities more precisely.

Probability Calculation

Probability calculation is the core of analyzing events in probability. The probabilities assigned to events help us understand their likelihood under given conditions.

For Event A (even-numbered months), the calculation involves identifying the even months: February, April, June, August, October, and December. Since there are 6 even months, the probability becomes:

• \( P(A) = \frac{6}{12} = \frac{1}{2} \)

Similar calculations apply to other events. For instance, with Event B1 (first half of the year), all months from January to June contribute to its probability:

• \( P(B1) = \frac{6}{12} = \frac{1}{2} \)

Furthermore, to determine whether events are independent, we can test if the probability of their intersection equals the product of their probabilities. For example:

• If \( P(A \cap B1) = P(A) \times P(B1) \)

This equation's validity confirms independence. If valid, the events do not affect each other's probabilities. These calculations are essential for drawing connections and conclusions in probability problems. Always consider both individual and combined event probabilities for a complete analysis.