Let's dive into understanding the concept of conditional probability to determine the correct answer for the given question. We'll go through each option one by one:
### Option A
- Finding the probability of two independent events occurring at the same time
- This scenario describes the probability of the intersection of two independent events, often denoted as [tex]\( P(A \cap B) = P(A) \times P(B) \)[/tex].
- This is not the definition of conditional probability but rather the joint probability of two independent events.
### Option B
- Finding the probability of an event occurring two or more times
- This option could be related to repeated trials or the Poisson distribution, which deals with the probability of a given number of events happening in a fixed interval of time or space.
- Again, this is not the concept of conditional probability.
### Option C
- Finding the probability of an event occurring only once
- This refers to a simple probability scenario where we are considering the probability of a single event occurring.
- This does not account for the dependence on another event, which is crucial in conditional probability.
### Option D
- Finding the probability of an event occurring given another event had already occurred
- This is the precise definition of conditional probability.
- Conditional probability is denoted as [tex]\( P(A|B) \)[/tex], which means the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has already occurred. It is calculated using the formula:
[tex]\[
P(A|B) = \frac{P(A \cap B)}{P(B)}
\][/tex]
- This explicitly describes a situation where the occurrence of one event influences the probability of another event.
### Conclusion
By examining all the options, it is clear that the situation which best describes conditional probability is:
- Option D: finding the probability of an event occurring given another event had already occurred
So, the correct answer is:
D
