91Ó°ÊÓ

In probability, it's important to understand the concept of independent events. Two events are said to be independent if the occurrence of one does not affect the probability of the other occurring.
For example, tossing a coin and rolling a dice are independent since the result of the coin does not influence the dice roll.
To determine if events are independent, you can use this rule:

• If event A and event B are independent, then the probability of both A and B occurring is the product of their individual probabilities:

\[ P(A \text{ and } B) = P(A) \times P(B) \]In the exercise given, events E and F are independent. Therefore, the conditional probability \(P(E \mid F)\) is simply \(P(E)\). This simplification is because F happening does not change how likely E is to occur.

Conditional probability is a measure of how likely an event is to occur given that another event has already occurred.
It's denoted as \(P(A \mid B)\), read as the probability of A given B. It’s different from regular probability since it's based on an additional condition.
To calculate conditional probability, you can use the formula:

• \[ P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)} \]

However, with independent events, conditional probability simplifies further. That's because \(P(A \mid B) = P(A)\). The additional condition doesn’t alter the likelihood since the events don’t influence each other.
In the problem, given that \(P(E \mid F) = .36\) and E and F are independent, this directly means \(P(E) = .36\), as F happening doesn't modify how probable E is.

Mathematical notation is like a language to express mathematical ideas clearly and concisely. In probability, understanding the symbols and notations is crucial.
Here are some key notations used in the exercise:

• \(P(X)\): Probability of event X occurring.
• \(P(A \text{ and } B)\): Probability of both A and B occurring together.
• \(P(A \mid B)\): Conditional probability of A given B has occurred.

These symbols compactly convey the relationships between probabilities and conditions. Understanding them not only helps in applying formulas correctly but also aids in efficiently solving problems.
In the given exercise, the notation \(P(E \mid F) = .36\) succinctly represents the probability of E happening if F is already known to occur. Using the concept that E and F are independent, we further deduced \(P(E)\) directly from this notation.