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enter image description here You visit your grandparents and notice an old die and notebook. The die is handmade and the edges are worn. Grandpa explains that he rolled this die 10,000 times, independently and under the same conditions, when he was a prisoner of war and recorded the results of the die rolls in his notebook:

  • Side of the die 1 2 3 4 5 6
  • Number of outcomes 2607 1633 1148 1839 2552 221

He adds that he used to run an illegal gambling ring and that he fashioned this die out of bone himself. Which of the following follow from these observations? I. The data suggests the die has a much lower probability of producing a ‘6’ compared to other outcomes. II. It is possible, though very unlikely, that the die is fair.

How do I determine if the die is fair? The P(6) here is 0.0221 and if I'm not wrong, both options I and II are correct?

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    $\begingroup$ You are correct. Although, it is extremely unlikely the die is fair. Have you studied hypothesis testing? You could use the null hypothesis $$H_0: \text{The probability of rolling a } 6 \text{ is } \frac{1}{6},$$ and use the $p$-value to support both statements. $\endgroup$ – David Feb 9 '17 at 6:33
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    $\begingroup$ No matter what you got, it is always possible that the die is fair. Even if you roll 6 a million times in a row, you cannot exclude the possibility that the die is fair; it's just extremely unlikely. So it's proposition I that has to be tested: Is the data already far enough from an equal distribution that it is not to be expected from a fair die? $\endgroup$ – celtschk Feb 9 '17 at 6:39
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The chi-squared goodness-of-fit test, for this situation uses the test statistic $$Q = \sum_{i=1}^6 \frac{(X_i - E)^2}{E},$$ where the $X_i$ are the observed counts 2607, 1633, and so on; and $E = np = 10000\left(\frac{1}{6}\right),$ for a fair die.

The statistic $Q$ is approximately distributed as $\mathsf{Chisq}(6-1=5),$ so you would reject (5% level) the null hypothesis that the die is fair if $Q > 11.07,$ which you can get from a printed table of the chi-squared distribution or from software.

Small values of $Q$ indicate a good fit of the observed values to the expected values, and large values indicate a bad fit. In your case, $Q = 2434.7,$ which is hugely larger than 11.07---an extremely bad fit. The P-value of the test is so small that it computes as "$0$". So it is extremely unlikely that a fair die would give such a large value of $Q$. The trouble, of course, is that far too few 6's were observed.

Anything is "possible," of course. But you could have started doing 10,000-roll experiments at the time of the "Big Bang" with a very durable fair die without getting results as extreme as these. So I'd say this is grandpa's crooked die.

Some computations are shown below using R statistical software, but you could compute $Q$ using a calculator.

qchisq(.95,5)
## 11.0705                       
X = c(2607, 1633, 1148, 1839, 2552, 221)
E = 10000/6
Q = sum((X-E)^2/E);  Q
## 2434.705
p.val = 1 - pchisq(Q, 5);  p.val
## 0
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