Question
EXPLAIN how to determine theoretical probability of an event.
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Answered by
GPT-4o mini
Theoretical probability is determined by analyzing the set of all possible outcomes in a given scenario and identifying how many of those outcomes correspond to a specific event of interest. Here are the steps to calculate theoretical probability:
1. **Define the Experiment**: Clearly identify the experiment you are considering and what the outcomes will be. For example, if you're rolling a die, the experiment is rolling the die.
2. **Identify the Sample Space (S)**: The sample space is the set of all possible outcomes of the experiment. For a six-sided die, the sample space would be {1, 2, 3, 4, 5, 6}.
3. **Identify the Event (E)**: Define the specific event you want to calculate the probability for. For example, if you're interested in rolling an even number, your event would be {2, 4, 6}.
4. **Count the Outcomes**:
- Determine the total number of outcomes in the sample space (n(S)). For our die example, there are 6 possible outcomes (1, 2, 3, 4, 5, 6).
- Count the number of favourable outcomes for the event (n(E)). In our case, there are 3 favourable outcomes for rolling an even number (2, 4, 6).
5. **Apply the Probability Formula**: The theoretical probability (P) of the event can be calculated using the formula:
\[
P(E) = \frac{n(E)}{n(S)}
\]
where:
- \( P(E) \) is the probability of the event E,
- \( n(E) \) is the number of favourable outcomes for event E,
- \( n(S) \) is the total number of outcomes in the sample space.
6. **Calculate the Probability**: Using the numbers from our example,
- Total outcomes \( n(S) = 6 \)
- Favourable outcomes \( n(E) = 3 \)
The probability of rolling an even number is:
\[
P(E) = \frac{3}{6} = \frac{1}{2}
\]
Thus, the theoretical probability of rolling an even number on a six-sided die is \(\frac{1}{2}\) or 0.5.
By following these steps, you can determine the theoretical probability of any event given a well-defined experiment.
1. **Define the Experiment**: Clearly identify the experiment you are considering and what the outcomes will be. For example, if you're rolling a die, the experiment is rolling the die.
2. **Identify the Sample Space (S)**: The sample space is the set of all possible outcomes of the experiment. For a six-sided die, the sample space would be {1, 2, 3, 4, 5, 6}.
3. **Identify the Event (E)**: Define the specific event you want to calculate the probability for. For example, if you're interested in rolling an even number, your event would be {2, 4, 6}.
4. **Count the Outcomes**:
- Determine the total number of outcomes in the sample space (n(S)). For our die example, there are 6 possible outcomes (1, 2, 3, 4, 5, 6).
- Count the number of favourable outcomes for the event (n(E)). In our case, there are 3 favourable outcomes for rolling an even number (2, 4, 6).
5. **Apply the Probability Formula**: The theoretical probability (P) of the event can be calculated using the formula:
\[
P(E) = \frac{n(E)}{n(S)}
\]
where:
- \( P(E) \) is the probability of the event E,
- \( n(E) \) is the number of favourable outcomes for event E,
- \( n(S) \) is the total number of outcomes in the sample space.
6. **Calculate the Probability**: Using the numbers from our example,
- Total outcomes \( n(S) = 6 \)
- Favourable outcomes \( n(E) = 3 \)
The probability of rolling an even number is:
\[
P(E) = \frac{3}{6} = \frac{1}{2}
\]
Thus, the theoretical probability of rolling an even number on a six-sided die is \(\frac{1}{2}\) or 0.5.
By following these steps, you can determine the theoretical probability of any event given a well-defined experiment.