A Die Is Thrown 120 Times. What Is The Expected Frequency Of Rolling A Number Greater Than 4?
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 rolling 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 rolling a number greater than 4, which means we want to count the number of times the die lands on 5 or 6.
Theoretical Background
The probability of rolling a number greater than 4 on a single throw of a die is 2/6, since there are two favorable outcomes (5 and 6) out of a total of six possible outcomes. This probability can be written as:
P(X > 4) = 2/6 = 1/3
where X is the random variable representing the outcome of a single throw of the die.
Expected Frequency
The expected frequency of rolling a number greater than 4 in 120 throws of the die can be calculated using the formula:
E(X) = n * P(X > 4)
where n is the number of trials (120 in this case) and P(X > 4) is the probability of rolling a number greater than 4 (1/3).
Plugging in the values, we get:
E(X) = 120 * 1/3 = 40
So, the expected frequency of rolling a number greater than 4 in 120 throws of the die is 40.
Standard Deviation
The standard deviation of the expected frequency is a measure of the variability or dispersion of the expected frequency. It can be calculated using the formula:
σ = √(n * P(X > 4) * (1 - P(X > 4)))
where σ is the standard deviation and n is the number of trials.
Plugging in the values, we get:
σ = √(120 * 1/3 * (1 - 1/3)) = √(120 * 1/3 * 2/3) = √(80/9) ≈ 4.62
So, the standard deviation of the expected frequency is approximately 4.62.
Interpretation
The expected frequency of rolling a number greater than 4 in 120 throws of the die is 40. This means that, on average, we expect to roll a number greater than 4 approximately 40 times in 120 throws. The standard deviation of the expected frequency is approximately 4.62, which means that the actual frequency of rolling a number greater than 4 may vary by up to 4.62 times the expected frequency.
Conclusion
In conclusion, the expected frequency of rolling a number greater than 4 in 120 throws of the die is 40. The standard deviation of the expected frequency is approximately 4.62. This means that, on average, we expect to roll a number greater than 4 approximately 40 times in 120 throws, with a possible variation of up to 4.62 times the expected frequency.
Frequently Asked Questions
Q: What is the probability of rolling a number greater than 4 on a single throw of a die?
A: The probability of rolling a number greater than 4 on a single throw of a die is 2/6, which is equal to 1/3.
Q: What is the expected frequency of rolling a number greater than 4 in 120 throws of the die?
A: The expected frequency of rolling a number greater than 4 in 120 throws of the die is 40.
Q: What is the standard deviation of the expected frequency?
A: The standard deviation of the expected frequency is approximately 4.62.
Q: What does the expected frequency represent?
A: The expected frequency represents the average number of times the event (rolling a number greater than 4) is expected to occur in a given number of trials (120 throws of the die).
Q: What does the standard deviation represent?
A: The standard deviation represents the variability or dispersion of the expected frequency.
References
- [1] Probability Theory and Statistics, by William Feller
- [2] Introduction to Probability and Statistics, by William Feller
- [3] Probability and Statistics, by James E. Gentle
Glossary
- Expected frequency: The average number of times an event is expected to occur in a given number of trials.
- Standard deviation: A measure of the variability or dispersion of the expected frequency.
- Probability: A measure of the likelihood of an event occurring.
- Random variable: A variable that takes on a value randomly.
- Trial: A single attempt to observe the outcome of an event.
A Die is Thrown 120 Times: Expected Frequency of Rolling a Number Greater than 4 - Q&A ====================================================================================
Introduction
In our previous article, we explored the expected frequency of rolling 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 number greater than 4 on a single throw of a die?
A: The probability of rolling a number greater than 4 on a single throw of a die is 2/6, which is equal to 1/3.
Q: What is the expected frequency of rolling a number greater than 4 in 120 throws of the die?
A: The expected frequency of rolling a number greater than 4 in 120 throws of the die is 40.
Q: What is the standard deviation of the expected frequency?
A: The standard deviation of the expected frequency is approximately 4.62.
Q: What does the expected frequency represent?
A: The expected frequency represents the average number of times the event (rolling a number greater than 4) is expected to occur in a given number of trials (120 throws of the die).
Q: What does the standard deviation represent?
A: The standard deviation represents the variability or dispersion of the expected frequency.
Q: How is the expected frequency calculated?
A: The expected frequency is calculated using the formula:
E(X) = n * P(X > 4)
where n is the number of trials (120 in this case) and P(X > 4) is the probability of rolling a number greater than 4 (1/3).
Q: What is the significance of the standard deviation?
A: The standard deviation is a measure of the variability or dispersion of the expected frequency. It can be used to estimate the range of possible values for the expected frequency.
Q: Can the expected frequency be used to make predictions?
A: Yes, the expected frequency can be used to make predictions about the number of times a particular event will occur in a given number of trials.
Q: What are some common applications of expected frequency?
A: Expected frequency is used in a variety of fields, including statistics, probability, and engineering. It is used to make predictions, estimate variability, and understand the behavior of random systems.
Q: How can the expected frequency be used in real-world scenarios?
A: The expected frequency can be used in a variety of real-world scenarios, such as:
- Predicting the number of customers who will visit a store in a given day
- Estimating the number of defects in a manufacturing process
- Understanding the behavior of a random system, such as a stock market or a weather pattern
Conclusion
In conclusion, the expected frequency of rolling a number greater than 4 in 120 throws of the die is 40. The standard deviation of the expected frequency is approximately 4.62. The expected frequency can be used to make predictions, estimate variability, and understand the behavior of random systems.
Frequently Asked Questions (FAQs)
Q: What is the probability of rolling a number greater than 4 on a single throw of a die?
A: The probability of rolling a number greater than 4 on a single throw of a die is 2/6, which is equal to 1/3.
Q: What is the expected frequency of rolling a number greater than 4 in 120 throws of the die?
A: The expected frequency of rolling a number greater than 4 in 120 throws of the die is 40.
Q: What is the standard deviation of the expected frequency?
A: The standard deviation of the expected frequency is approximately 4.62.
Q: What does the expected frequency represent?
A: The expected frequency represents the average number of times the event (rolling a number greater than 4) is expected to occur in a given number of trials (120 throws of the die).
Q: What does the standard deviation represent?
A: The standard deviation represents the variability or dispersion of the expected frequency.
References
- [1] Probability Theory and Statistics, by William Feller
- [2] Introduction to Probability and Statistics, by William Feller
- [3] Probability and Statistics, by James E. Gentle
Glossary
- Expected frequency: The average number of times an event is expected to occur in a given number of trials.
- Standard deviation: A measure of the variability or dispersion of the expected frequency.
- Probability: A measure of the likelihood of an event occurring.
- Random variable: A variable that takes on a value randomly.
- Trial: A single attempt to observe the outcome of an event.