A regular die has six sides with numbers 1 to 6 marked on its sides. If a very large number of throws show the following frequencies of occurrence:
1 → 0.167; 2 → 0.167; 3 → 0.152; 4 → 0.166; 5 → 0.168; 6 → 0.180. We call this die
- Airregular
- Bbiased
- CGaussian
- Dinsufficient
Solution & Step-by-step Explanation
Die Probabilities and Observed Frequencies: Comparing Expected vs. Actual
A standard, fair six-sided die has numbers 1 to 6 marked on its sides. For a fair die, each face has an equal chance of landing face up after a throw. Theoretically, the probability of any specific number appearing is the same for all numbers. Since there are six possible outcomes (1, 2, 3, 4, 5, 6), the theoretical probability for each outcome is:
To understand this value better, we can convert the fraction to a decimal:
This means that if we throw a fair die a very large number of times, we expect the frequency of each number appearing to be approximately 0.1667.
The question provides the observed frequencies from a very large number of throws for a specific die:
- 1 → 0.167
- 2 → 0.167
- 3 → 0.152
- 4 → 0.166
- 5 → 0.168
- 6 → 0.180
Let's compare these observed frequencies to the theoretical probability of 0.1667:
Analyzing Die Frequencies: Identifying Bias
Observing the table, we can see that the frequencies for faces 1, 2, 4, and 5 are quite close to the theoretical probability of 0.1667. However, the frequencies for face 3 (0.152) and face 6 (0.180) show noticeable deviations. The frequency for face 3 is lower than expected, while the frequency for face 6 is higher than expected.
When the observed frequencies of outcomes from repeated trials (like throwing a die) consistently differ from the expected probabilities of a fair process, we say that the process or the object (in this case, the die) is biased.
A biased die means that the die is weighted or shaped in such a way that certain outcomes are more likely to occur than others, deviating from the equal probability assumption of a fair die.
Understanding Statistical Terms: Biased vs. Irregular vs. Gaussian
Let's consider the given options:
- Irregular: While the frequencies are not perfectly uniform, "irregular" is a general term. In statistics, when probabilities deviate systematically from fairness, a more specific term is used.
- Biased: This term accurately describes a die where the probabilities of outcomes are unequal due to physical imperfections or weighting. The observed frequencies strongly suggest this is the case, especially for faces 3 and 6.
- Gaussian: A Gaussian (or normal) distribution describes a bell-shaped curve, typically for continuous data. Dice throws are discrete events, and their probabilities don't necessarily follow a Gaussian pattern. This option is not relevant here.
- Insufficient: The question explicitly states "a very large number of throws," indicating that the data collected is sufficient to observe the die's behavior and make a conclusion about its fairness.
Based on the analysis of the observed frequencies, which deviate from the theoretical probabilities expected for a fair die, the most appropriate term to describe this die is biased.
A standard, fair six-sided die has numbers 1 to 6 marked on its sides. For a fair die, each face has an equal chance of landing face up after a throw. Theoretically, the probability of any specific number appearing is the same for all numbers. Since there are six possible outcomes (1, 2, 3, 4, 5, 6), the theoretical probability for each outcome is:
To understand this value better, we can convert the fraction to a decimal:
This means that if we throw a fair die a very large number of times, we expect the frequency of each number appearing to be approximately 0.1667.
The question provides the observed frequencies from a very large number of throws for a specific die:
- 1 → 0.167
- 2 → 0.167
- 3 → 0.152
- 4 → 0.166
- 5 → 0.168
- 6 → 0.180
Let's compare these observed frequencies to the theoretical probability of 0.1667:
| Face Number | Observed Frequency | Theoretical Probability (1/6) | Difference from Theoretical |
|---|---|---|---|
| 1 | 0.167 | 0.1667 | |
| 2 | 0.167 | 0.1667 | |
| 3 | 0.152 | 0.1667 | |
| 4 | 0.166 | 0.1667 | |
| 5 | 0.168 | 0.1667 | |
| 6 | 0.180 | 0.1667 |
Observing the table, we can see that the frequencies for faces 1, 2, 4, and 5 are quite close to the theoretical probability of 0.1667. However, the frequencies for face 3 (0.152) and face 6 (0.180) show noticeable deviations. The frequency for face 3 is lower than expected, while the frequency for face 6 is higher than expected.
When the observed frequencies of outcomes from repeated trials (like throwing a die) consistently differ from the expected probabilities of a fair process, we say that the process or the object (in this case, the die) is biased.
A biased die means that the die is weighted or shaped in such a way that certain outcomes are more likely to occur than others, deviating from the equal probability assumption of a fair die.
Understanding Statistical Terms: Biased vs. Irregular vs. Gaussian
Let's consider the given options:
- Irregular: While the frequencies are not perfectly uniform, "irregular" is a general term. In statistics, when probabilities deviate systematically from fairness, a more specific term is used.
- Biased: This term accurately describes a die where the probabilities of outcomes are unequal due to physical imperfections or weighting. The observed frequencies strongly suggest this is the case, especially for faces 3 and 6.
- Gaussian: A Gaussian (or normal) distribution describes a bell-shaped curve, typically for continuous data. Dice throws are discrete events, and their probabilities don't necessarily follow a Gaussian pattern. This option is not relevant here.
- Insufficient: The question explicitly states "a very large number of throws," indicating that the data collected is sufficient to observe the die's behavior and make a conclusion about its fairness.
Based on the analysis of the observed frequencies, which deviate from the theoretical probabilities expected for a fair die, the most appropriate term to describe this die is biased.