Homework 4

Statistical Independence

Statistical independence is a key concept in probability and statistics, describing a relationship between two events (or variables) where the occurrence of one event does not affect the occurrence of the other.

Defining Statistical Independence

Two events, say \(A\) and \(B\), are considered independent if the occurrence of one does not influence the likelihood of the other happening. Formally, we can define this independence using probabilities: events \(A\) and \(B\) are independent if and only if:

\[P(A \cap B) = P(A) \cdot P(B)\]

where:

• \(P(A \cap B)\) is the probability that both events \(A\) and \(B\) occur together.
• \(P(A)\) and \(P(B)\) are the individual probabilities of \(A\) and \(B\) occurring.

Understanding the Formula

The formula \(P(A \cap B) = P(A) \cdot P(B)\) implies that if \(A\) and \(B\) are independent, the likelihood of both events occurring together is simply the product of their individual probabilities. This means that knowing one event has happened does not provide any information about the probability of the other event happening.

Independence Through Conditional Probability

Another way to define independence is with conditional probability. The conditional probability of \(A\) given \(B\) , written \(P(A | B)\) , is the probability of \(A\) occurring given that \(B\) has already occurred. For \(A\) and \(B\) to be independent, the occurrence of \(B\) should not change the probability of \(A\) . Therefore, if \(A\) and \(B\) are independent:

\[P(A | B) = P(A)\]

This means that knowing \(B\) has occurred does not affect the likelihood of \(A\), reinforcing that \(A\) and \(B\) are independent.

Comparing to Dependent Events

In contrast, if two events are dependent, the occurrence of one event affects the probability of the other. For dependent events \(A\) and \(B\), we would observe that \(P(A \cap B) \neq P(A) \cdot P(B)\) and consequently, \(P(A | B) \neq P(A)\) .

Example

Imagine rolling two six-sided dice, one green and one red. Let:

• Event \(A\) be “the green die lands on 4.”
• Event \(B\) be “the red die lands on 6.”

The probability of \(A\) (the green die landing on 4) is \(\frac{1}{6}\), and the probability of \(B\) (the red die landing on 6) is also \(\frac{1}{6}\). Because the outcome of one die does not influence the outcome of the other, the probability of both events happening together is simply:

\[P(A \cap B) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\]

This shows that the events are independent: knowing that one die shows a particular number doesn’t tell us anything about the other.

Summary

These examples highlight statistical independence in different contexts, demonstrating that the occurrence of one independent event does not impact the probability of another.