A Student Believes That A Certain Number Cube Is Unfair And Is More Likely To Land With A Six Facing Up. The Student Rolls The Number Cube 45 Times, And The Cube Lands With A Six Facing Up 12 Times. Assuming The Conditions For Inference Have Been Met,
**A Student's Dilemma: Is the Number Cube Unfair?**
Understanding the Problem
A student has raised concerns that a certain number cube is unfair, suggesting that it is more likely to land with a six facing up. To investigate this claim, the student rolls the number cube 45 times and observes that it lands with a six facing up 12 times. In this article, we will delve into the world of probability and statistics to determine whether the student's suspicions are justified.
The Conditions for Inference
Before we begin our analysis, it is essential to ensure that the conditions for inference have been met. These conditions are:
- Independence: Each roll of the number cube is independent of the others.
- Randomness: The rolls are truly random, and the outcome of each roll is not influenced by external factors.
- Large Sample Size: The student has rolled the number cube a sufficient number of times (45) to make inferences about the probability of landing on a six.
Assuming these conditions have been met, we can proceed with our analysis.
Calculating the Probability
To determine whether the number cube is unfair, we need to calculate the probability of landing on a six. The probability of an event is defined as the number of favorable outcomes divided by the total number of possible outcomes.
In this case, the number of favorable outcomes is the number of times the cube lands on a six (12), and the total number of possible outcomes is the total number of rolls (45).
Probability Formula
The probability of landing on a six is calculated as follows:
P(six) = Number of favorable outcomes / Total number of possible outcomes = 12 / 45 = 0.2667
Interpreting the Results
The calculated probability of 0.2667 suggests that the number cube is not significantly biased towards landing on a six. In fact, the probability of landing on a six is approximately 26.67%, which is close to the expected probability of 1/6 (16.67%) for a fair number cube.
Conclusion
Based on the analysis, it appears that the number cube is not unfair and is not more likely to land on a six. The student's suspicions were likely due to chance, and the observed frequency of landing on a six is consistent with the expected probability for a fair number cube.
Frequently Asked Questions
Q: What is the probability of landing on a six if the number cube is fair?
A: The probability of landing on a six for a fair number cube is 1/6, which is approximately 0.1667.
Q: How many times did the student roll the number cube?
A: The student rolled the number cube 45 times.
Q: What is the expected frequency of landing on a six for a fair number cube?
A: The expected frequency of landing on a six for a fair number cube is 45 x 1/6 = 7.5.
Q: Is the number cube biased towards landing on a six?
A: No, the analysis suggests that the number cube is not biased towards landing on a six.
Q: What are the conditions for inference in this problem?
A: The conditions for inference are independence, randomness, and a large sample size.
Q: How do we calculate the probability of an event?
A: The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes.
Q: What is the probability of landing on a six in this problem?
A: The probability of landing on a six is 0.2667.
Q: Is the observed frequency of landing on a six consistent with the expected probability for a fair number cube?
A: Yes, the observed frequency of landing on a six is consistent with the expected probability for a fair number cube.