Question

Isabella is studying the fairness of a six-sided numbered cube with numbers 1,2,3,4,5, and 6.

The numbered cube is rolled 36 times, and the numbers on the top side are recorded in the table below.

1 4 1 3 6 2

3 3 5 6 1 6

2 5 4 6 4 2

5 6 1 5 3 6

3 4 3 6 4 1

1 2 5 2 5 3



Based on this data, what conclusion would you make about the fairness of the numbered cube?
Justify your answer.

Answer 1

Count the outcomes (36 rolls):

- 1: 6 times
- 2: 5 times
- 3: 7 times
- 4: 5 times
- 5: 6 times
- 6: 7 times

If the cube is fair, each face should appear 36/6 = 6 times on average. The observed counts deviate from 6 by at most 1 for each face. A chi-square goodness-of-fit statistic is

chi2 = sum (O−E)^2/E = (0^2+(-1)^2+(1)^2+(-1)^2+0^2+(1)^2)/6 = 4/6 ≈ 0.667.

With 5 degrees of freedom the critical chi-square at the 5% level is about 11.07, and 0.667 << 11.07. Thus we fail to reject the null hypothesis of fairness. Conclusion: the data give no evidence that the cube is unfair — the results are consistent with a fair die.