91Ó°ÊÓ

In probability, two events are considered independent if the occurrence of one event does not affect the likelihood of the other event happening. To check for independence, we use the formula:

• Probability of both events occurring: \( P(A \cap B) \)
• Product of individual probabilities: \( P(A) \times P(B) \)

If \( P(A \cap B) = P(A) \times P(B) \), the events are independent.
Tackling the exercise example:

• Probability of choosing a female: \( \frac{65}{142} \)
• Probability of running a first 5K: \( \frac{47}{142} \)
• Probability of a female running her first 5K: \( \frac{19}{142} \)

Multiply the probabilities of being female and running a first 5K:
\[ \frac{65}{142} \times \frac{47}{142} = \frac{3055}{20164} \]
Compare with the probability of being a female running a first 5K, \( \frac{19}{142} \), which simplifies to \( \frac{2717}{20164} \).
Since they are not equal, the events are not independent.

Events are mutually exclusive if the occurrence of one event means the other cannot happen at the same time. Simply put, both events cannot occur simultaneously.
In this exercise, we assess the events "being female" and "participating in their first 5K road race" to see if they are mutually exclusive.
If these events were mutually exclusive, then a female runner participating in her first 5K would be impossible. However, the data shows there are 19 females running their first 5K.
These two facts prove that these events are not mutually exclusive, as both can and do occur at the same time for some of the participants.

Conditional probability helps us understand the probability of an event happening, given that another event has already occurred. The formula for conditional probability is:

• \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)

Here, \( P(A \mid B) \) is the probability of event A happening given event B is true.
Referring to the given scenario, let's calculate the probability a participant is participating in their first 5K, given they are female:

• Probability a participant is female and running their first 5K: \( \frac{19}{142} \)
• Probability a participant is female: \( \frac{65}{142} \)

So, \( P(\text{First 5K} \mid \text{Female}) = \frac{\frac{19}{142}}{\frac{65}{142}} = \frac{19}{65} \).
This tells us, given a runner is female, the likelihood she is running her first 5K is \( \frac{19}{65} \), showcasing how event occurrence affects probabilities with conditional probability.