Expected Frequency Formula

Oct 13, 2025 by windailys.com 18 views

Introduction

In probability theory, the expected frequency of an event is the average number of times the event is expected to occur in a given number of trials. In this article, we will explore the expected frequency of a number greater than 4 when a die is thrown 120 times.

Understanding the Problem

When a die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, and 6. We are interested in finding the expected frequency of numbers greater than 4, which means we want to count the number of times the die shows a 5 or a 6.

Theoretical Background

The probability of an event occurring is defined as the number of favorable outcomes divided by the total number of possible outcomes. In this case, the probability of rolling a 5 or a 6 is 2/6, since there are two favorable outcomes (5 and 6) out of a total of six possible outcomes.

Expected Frequency Formula

The expected frequency of an event is calculated using the formula:

Expected Frequency = (Number of Trials) x (Probability of the Event)

In this case, the number of trials is 120 (the number of times the die is thrown), and the probability of rolling a 5 or a 6 is 2/6.

Calculating the Expected Frequency

Using the formula above, we can calculate the expected frequency of a number greater than 4 as follows:

Expected Frequency = (120) x (2/6) Expected Frequency = 40

Interpretation of Results

The expected frequency of a number greater than 4 is 40. This means that, on average, we expect to see a 5 or a 6 approximately 40 times when a die is thrown 120 times.

Standard Deviation and Variance

To understand the variability of the expected frequency, we need to calculate the standard deviation and variance. The standard deviation is a measure of the amount of variation or dispersion from the expected value, while the variance is the average of the squared differences from the expected value.

Calculating the Standard Deviation and Variance

The standard deviation and variance can be calculated using the following formulas:

Standard Deviation = √(Variance) Variance = (Expected Frequency) x (1 - (Probability of the Event))

Using the formulas above, we can calculate the standard deviation and variance as follows:

Variance = (40) x (1 - (2/6)) Variance = 20 Standard Deviation = √(20) Standard Deviation = 4.47

Interpretation of Standard Deviation and Variance

The standard deviation of 4.47 indicates that the expected frequency of a number greater than 4 is likely to vary by approximately 4.47 times the expected value. The variance of 20 indicates that the squared differences from the expected value are, on average, 20.

Conclusion

In conclusion, the expected frequency of a number greater than 4 when a die is thrown 120 times is 40. This means that, on average, we expect to see a 5 or a 6 approximately 40 times. The standard deviation and variance provide a measure of the variability of the expected frequency, indicating that the expected frequency is likely to vary by approximately 4.47 times the expected value.

References

[1] Probability Theory and Statistics, by William Feller
[2] Statistics for Dummies, by Deborah J. Rumsey

Frequently Asked Questions

Q: What is the expected frequency of a number greater than 4 when a die is thrown 120 times? A: The expected frequency is 40.
Q: What is the standard deviation of the expected frequency? A: The standard deviation is 4.47.
Q: What is the variance of the expected frequency? A: The variance is 20.
A Die is Thrown 120 Times: Expected Frequency of a Number Greater than 4 - Q&A ====================================================================

Introduction

In our previous article, we explored the expected frequency of a number greater than 4 when a die is thrown 120 times. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the probability of rolling a 5 or a 6 on a die?

A: The probability of rolling a 5 or a 6 on a die is 2/6, since there are two favorable outcomes (5 and 6) out of a total of six possible outcomes.

Q: How is the expected frequency calculated?

A: The expected frequency is calculated using the formula:

Expected Frequency = (Number of Trials) x (Probability of the Event)

In this case, the number of trials is 120 (the number of times the die is thrown), and the probability of rolling a 5 or a 6 is 2/6.

Q: What is the expected frequency of a number greater than 4 when a die is thrown 120 times?

A: The expected frequency of a number greater than 4 is 40.

Q: What is the standard deviation of the expected frequency?

A: The standard deviation of the expected frequency is 4.47.

Q: What is the variance of the expected frequency?

A: The variance of the expected frequency is 20.

Q: How does the expected frequency relate to the actual frequency?

A: The expected frequency is an average value that we expect to see in a large number of trials. The actual frequency may vary from the expected frequency due to random fluctuations.

Q: Can we use the expected frequency to make predictions about future outcomes?

A: Yes, we can use the expected frequency to make predictions about future outcomes. However, it's essential to note that the expected frequency is an average value, and actual outcomes may vary.

Q: What are some real-world applications of expected frequency?

A: Expected frequency has many real-world applications, such as:

Predicting the number of customers who will visit a store on a given day
Estimating the number of people who will attend a concert or event
Forecasting the number of sales for a product

Q: How can we use expected frequency to make informed decisions?

A: Expected frequency can be used to make informed decisions by:

Identifying areas where we can improve our processes or products
Allocating resources more effectively
Making predictions about future outcomes

Conclusion

In conclusion, the expected frequency of a number greater than 4 when a die is thrown 120 times is 40. We have also answered some frequently asked questions related to the topic, including the probability of rolling a 5 or a 6, the calculation of expected frequency, and the standard deviation and variance of the expected frequency.

References

[1] Probability Theory and Statistics, by William Feller
[2] Statistics for Dummies, by Deborah J. Rumsey

Frequently Asked Questions

Q: What is the expected frequency of a number greater than 4 when a die is thrown 120 times? A: The expected frequency is 40.
Q: What is the standard deviation of the expected frequency? A: The standard deviation is 4.47.
Q: What is the variance of the expected frequency? A: The variance is 20.