91Ó°ÊÓ

In probability theory, events are considered independent if the occurrence of one event does not affect the probability of the other. When evaluating independence, we look at whether the joint probability of two events equals the product of their individual probabilities. In other terms, events \( H \) (having a home phone) and \( C \) (having a cell phone) are independent if:

• \( P(H \cap C) = P(H) \times P(C) \)

For the survey data, we had:

• \( P(H \cap C) = 0.58 \)
• \( P(H) = 0.73 \)
• \( P(C) = 0.83 \)

Calculating the product of \( P(H) \) and \( P(C) \):

• \( 0.73 \times 0.83 = 0.6059 \)

Since \( 0.58 eq 0.6059 \), the events are not independent. This implies having a home phone does somewhat influence whether you have a cell phone.

When events are mutually exclusive, it means they cannot happen at the same time. In probability terms, if events \( H \) and \( C \) are mutually exclusive, then their joint probability is zero, meaning:

• \( P(H \cap C) = 0 \)

In the given survey, \( P(H \cap C) = 0.58 \), showing that there is a significant overlap between those who have a home phone and a cell phone. Therefore, these events are not mutually exclusive.
This means it is possible for individuals to have both types of phones simultaneously. So, the occurrence of one does not exclude the possibility of the other. This is a common occurrence in real life, where technological dependence makes it practical for people to own multiple communication devices.

Calculating probabilities is a crucial skill in statistics that helps us understand real-world event occurrences. The conditional probability asks us to determine the likelihood of one event happening, given another has already occurred.

• For the probability of having a cell phone given someone has a home phone, we use \( P(C|H) = \frac{P(H \cap C)}{P(H)} \).

Here, \( P(H \cap C) = 0.58 \) and \( P(H) = 0.73 \), so we calculate:

• \( P(C|H) = \frac{0.58}{0.73} \approx 0.7959 \)

This tells us that there's about a 79.59% probability that someone with a home phone also possesses a cell phone.
Conditional probability is particularly useful in making predictions or decisions based on known information or past data trends. Understanding how to compute it is essential for analyzing situations where events are connected or follow one another.