Lesson 2.5: Independence of Events

Introduction

In this lesson, we will explore the concept of statistical independence and its importance in probability theory. We will cover the definition of statistical independence based on the multiplication law, learn how to test for independence between two events, distinguish between independent events and mutually exclusive events, identify the conditions for independence, and determine whether given events are independent, supporting our conclusions with logical reasoning.

Learning Objectives

• Understand the definition of statistical independence in terms of the multiplication law.
• Learn how to test whether two events are statistically independent.
• Differentiate between independent events and mutually exclusive events.
• State the condition necessary for two events to be statistically independent.
• Evaluate examples of events to determine their independence and justify the conclusions drawn.

Statistical Independence

Definition

Statistical independence occurs when the outcome of one event does not affect the outcome of another. Mathematically, two events $ A $ and $ B $ are considered independent if the following condition holds:

$$\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$$

This equation states that the probability of both events occurring together is equal to the product of their individual probabilities.

Worked Example 1

Let us consider an example: Suppose we flip a fair coin and roll a fair six-sided die. Let event $ A $ be the coin landing on heads, and event $ B $ be rolling a 3 on the die. We can find:

• The probability of getting heads from the coin flip, $\Pr(A) = \frac{1}{2}$.
• The probability of rolling a 3, $\Pr(B) = \frac{1}{6}$.

Now, we find the probability of both events occurring.

Since the coin flip does not affect the result of rolling the die, we can calculate the probability of both happening:

$$\Pr(A \cap B) = \Pr(A) \cdot \Pr(B) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$$

Thus, for these events, we have:

$$\Pr(A \cap B) = \frac{1}{12}$$

Since $\Pr(A \cap B)$ is indeed equal to $\Pr(A) \cdot \Pr(B)$, events $ A $ and $ B $ are independent.

Testing for Independence

To test whether two events are statistically independent, we can follow these steps:

1. Calculate the individual probabilities of each event: $\Pr(A)$ and $\Pr(B)$.
2. Calculate the joint probability of both events occurring together: $\Pr(A \cap B)$.
3. Compare the joint probability with the product of the individual probabilities. If they are equal, the events are independent.

Worked Example 2

Consider a deck of 52 playing cards. Let $ A $ be the event of drawing a heart, and $ B $ be the event of drawing a red card (hearts or diamonds). First, calculate:

• $\Pr(A)$: There are 13 hearts in a deck of 52 cards, so $\Pr(A) = \frac{13}{52} = \frac{1}{4}$.
• $\Pr(B)$: There are 26 red cards in total (13 hearts + 13 diamonds). Thus, $\Pr(B) = \frac{26}{52} = \frac{1}{2}$.
• Next, we calculate $\Pr(A \cap B)$, which is the probability of drawing a heart (which is included in red cards): $\Pr(A \cap B) = \Pr(A) = \frac{13}{52} = \frac{1}{4}$.

Now, check whether $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$:

$$\Pr(A) \cdot \Pr(B) = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8}$$

Since $\Pr(A \cap B) = \frac{1}{4} \neq \frac{1}{8}$, the events $ A $ and $ B $ are not independent.

Distinction Between Independent Events and Mutually Exclusive Events

Independent events are situations where the occurrence of one does not impact the occurrence of the other. Conversely, two events are mutually exclusive when they cannot occur simultaneously. If $ A $ and $ B $ are mutually exclusive, then:

$$\Pr(A \cap B) = 0$$

This distinction is critical because it helps in understanding how the events interact.

Common Misconceptions

1. Mutually Exclusive implies Independence: Just because two events cannot happen at the same time does not mean they are independent. For example, if $ A $ is the event of rolling a 1 on a die and $ B $ is the event of rolling a 2, these are mutually exclusive events, and the occurrence of one certainly affects the probability of the other (specifically, it makes it impossible).

1. Dependent Events Misunderstood: If the occurrence of one event influences or changes the probability of another event occurring, they are termed dependent events. A common error is assuming independence when the events appear unrelated at first glance.

Conditions for Independence

To formally establish whether two events $ A $ and $ B $ are independent, the following condition must be satisfied:

$$\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$$

If this equality holds true, $ A $ and $ B $ are independent; otherwise, they are dependent.

Worked Example 3

Consider drawing a single card from a standard deck of cards. Let:

• $ A $ = the event that the card drawn is a face card. There are 12 face cards in a deck of 52 cards. Therefore, $\Pr(A) = \frac{12}{52} = \frac{3}{13}$.
• $ B $ = the event that the card drawn is a red card. $\Pr(B) = \frac{26}{52} = \frac{1}{2}$.
• Now, we can check if drawing a face card and a red card is independent.

If we look closely, we note that there are 6 red face cards (3 in hearts and 3 in diamonds): $\Pr(A \cap B) = \frac{6}{52} = \frac{3}{26}$.

• Now check whether $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$:

$$\Pr(A) \cdot \Pr(B) = \frac{3}{13} \cdot \frac{1}{2} = \frac{3}{26}$$

Thus, this time they are independent events.

Conclusion

This lesson has introduced the concept of independence in probability, defining it in terms of the multiplication law and providing the necessary steps to test whether two events are independent. We also distinguished between independent and mutually exclusive events and clarified common misconceptions. Understanding these principles is crucial as we progress to more complex concepts in probability.

Study Notes

• Statistical independence of events $ A $ and $ B $ requires that $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$.
• Independent events do not affect each other, while mutually exclusive events cannot occur together.
• To test for independence, compare joint probability with the product of individual probabilities.
• Events are independent if their probabilities fulfill the multiplication law.
• Pay attention to contextual clues that explain interactions between events to avoid misconceptions.