91Ó°ÊÓ

In probability theory, determining whether events are independent is crucial. Two events, let's say E and F, are independent if the occurrence of one does not affect the occurrence of the other. This means the probability that event E happens is the same, regardless of whether event F occurs.

Mathematically, independence is expressed as:
\[ P(E \text{ and } F) = P(E) \times P(F) \].

Another way to check for independence is through conditional probability. If events E and F are independent, the conditional probability of E given F, denoted as \( P(E \text{ or } F) \), is equal to the probability of E. Therefore, if:
\[ P(E \text{ or } F) = P(E) \],
then the events are independent.

In our exercise, we are given \( P(E) = 0.6 \) and \( P(E \text{ or } F) = 0.34 \). Since \( P(E \text{ or } F) eq P(E) \), we can conclude that events E and F are not independent.

Conditional probability deals with the probability of an event occurring, given that another event has already happened. It is denoted as \( P(E \text{ or } F) \), which reads as the probability of event E occurring given that event F has occurred.

This can be calculated using the formula:
\[ P(E \text{ or } F) = \frac{P(E \text{ and } F)}{P(F)} \],
provided that \( P(F) eq 0 \). Understanding conditional probability is important in determining the relationship between events.

In our exercise, we are provided with \( P(E) = 0.6 \) and \( P(E \text{ or } F) = 0.34 \). By comparing \( P(E \text{ or } F) \) with \( P(E) \), we can analyze whether the occurrence of event F has an impact on the likelihood of event E. If they are equal, it indicates that event F does not influence event E, pointing towards independence.

But since 0.34 is not equal to 0.6, it demonstrates that F indeed affects E, thereby showing the dependence between these two events.

Comparing probabilities is a key tool in probability theory, especially for understanding the relationship between events. In the context of independence and conditional probability, comparing given probabilities helps determine whether events influence each other.

To ascertain if events E and F are independent, we compare \( P(E) \) and \( P(E \text{ or } F) \). If \( P(E \text{ or } F) = P(E) \), then E and F are independent. Conversely, if \( P(E \text{ or } F) eq P(E) \), the events are dependent.

In the exercise provided, we have:

• \( P(E) = 0.6 \)
• \( P(E \text{ or } F) = 0.34 \)

Clearly, \( 0.34 \) is not equal to \( 0.6 \). This result tells us that events E and F are not independent; event F affects the probability of event E happening.

This comparison is simple yet powerful in revealing the nature of the relationships between events, making it a fundamental step in solving many probability problems.