- The theoretical probability of event \(E\) is denoted by \(P(E)\).
- The formula is the ratio of favorable outcomes to total outcomes.
- Favorable outcomes are the number of outcomes in \(E\).
- The denominator is the total number of possible outcomes in the sample space.
- Final answer: The theoretical probability of event \(E\), denoted by \(\boxed{P(E)}\), is the \(\boxed{\text{number of outcomes in } E}\) divided by \(\boxed{\text{total number of possible outcomes}}\).
Explanation
- Analyzing the definition of theoretical probability
To define the theoretical probability of an event \(E\), we look at how many ways that specific event can happen relative to the total number of all possible outcomes. This concept is fundamental to probability theory and provides a way to quantify uncertainty in experiments where all outcomes are equally likely.
- Defining the objective
The objective is to accurately fill in the blanks in the statement: "The theoretical probability of event \(E\), denoted by \(\_\_\_\_\), is the \(\_\_\_\_\) divided by \(\_\_\_\_\)." We need to define the notation and the two components of the probability fraction.
- Formulating the solution plan
- Identify the standard mathematical notation for the probability of an event \(E\), which is \(P(E)\).
- Recognize that the numerator represents the number of favorable outcomes (the number of outcomes that result in event \(E\) occurring).
- Recognize that the denominator represents the total number of possible outcomes in the experiment's sample space.
- Construct the complete definition: The theoretical probability of event \(E\), denoted by \(P(E)\), is the number of outcomes in \(E\) divided by the total number of possible outcomes.
- Defining the components of the probability formula
The theoretical probability of event \(E\) measures how likely a specific outcome or set of outcomes is to occur.
- Notation: We represent the probability of event \(E\) as \(P(E)\).
- Numerator: This is the count of all outcomes that satisfy the conditions defined by event \(E\). We call these the "favorable outcomes."
- Denominator: This is the count of every possible outcome that could occur in the experiment. This set of all possible outcomes is called the "sample space."
Therefore, the completed statement is: The theoretical probability of event \(E\), denoted by \(P(E)\), is the number of outcomes in \(E\) divided by the total number of possible outcomes.
Examples
When you roll a fair six-sided die, the sample space is \(\{1, 2, 3, 4, 5, 6\}\), meaning there are \(6\) possible outcomes. If event \(E\) is "rolling an even number," the favorable outcomes are \(\{2, 4, 6\}\), so there are \(3\) outcomes in \(E\). Using the formula, the probability is \(P(E) = \frac{3}{6} = \frac{1}{2}\).