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I think that the term 'by chance' is not clearly defined as long as you do not have a specific hypothesis you want to test.

You could regard the full text as your population. The complete 'census' of all words resulted in the 'true' parameter $\theta=.27$, say.

Now you describe that you took a 'sample' of words, whose characteristic is page position (first word on each page) and you want to test the hypothesis, whether page position affects the probability of a word being unique.

Hence you want to test: $$H_0:\theta=.27$$ which is equivalent to asking whether the sample of words comes from the population of all words (your full text) or forms an own (sub-) population.

If we regard the 44 pages (words) as independent draws from a Bernoulli distribution, the number of positive outcomes $X$ is Binomial. Now we need

$$P(X \ge 33|H_0) \approx 4.68*10^{-11}$$

As you can verify using R pbinom(32,44,.27,lower.tail=FALSE). This probability is very small, so you can say with very low probability of error that observing 33 unique of 44 words was not caused by chance alone, if the null hypothesis was true. Hence, $\theta$ of the sub-population of words at the top of all pages seems to be different from your population $\theta$ of .27.

Put differently, position seems to have an impact on the probability of uniqueness. Only in a very small proportion of cases you would make an error when claiming this.

I think that the term 'by chance' is not clearly defined as long as you do not have a specific hypothesis you want to test.

You could regard the full text as your population. The complete 'census' of all words resulted in the 'true' parameter $\theta=.27$, say.

Now you describe that you took a 'sample' of words, whose characteristic is page position (first word on each page) and you want to test the hypothesis, whether page position affects the probability of a word being unique.

Hence you want to test: $$H_0:\theta=.27$$ which is equivalent to asking whether the sample of words comes from the population of all words (your full text) or forms an own (sub-) population.

If we regard the 44 pages (words) as independent draws from a Bernoulli distribution, the number of positive outcomes $X$ is Binomial. Now we need

$$P(X \ge 33|H_0) \approx 4.68*10^{-11}$$

As you can verify using R pbinom(32,44,.27,lower.tail=FALSE). This probability is very small, so you can say with very low probability of error that observing 33 unique of 44 words was not caused by chance, because if the null hypothesis was true the pobability of this event happening by chane alone would be very small. Hence, $\theta$ of the sub-population of words at the top of all pages seems to be different from your population $\theta$ of .27.

Put differently, position seems to have an impact on the probability of uniqueness. Only in a very small proportion of cases you would make an error when claiming this.

I think that the term 'by chance' is not clearly defined as long as you do not have a specific hypothesis you want to test.

You could regard the full text as your population. The complete 'census' of all words resulted in the 'true' parameter $\theta=.27$, say.

Now you describe that you took a 'sample' of words, whose characteristic is page position (first word on each page) and you want to test the hypothesis, whether page position affects the propbability of a word being unique.

Hence you want to test: $$H_0:\theta=.27$$ which is equivalent to asking whether the sample of words comes from the population of all words (your full text) or forms an own (sub-) population.

If we regard the 44 pages (words) as independent draws from a Bernoulli distribution, the number of positive outcomes $X$ is Binmoial. Now we need

$$P(X \ge 33|H_0) \approx 4.68*10^{-11}$$

As you can verify using R pbinom(32,44,.27,lower.tail=FALSE). This probability is very small, so you can say with very low probability of error that observing 33 unique of 44 words was not caused by chance alome, if the null hypothesis was true. Hence, $\theta$ of the sub-population of words at the top of all pages seems to be different from your population $\theta$ of .27.

Put differently, position seems to have an impact on the probablity of uniqueness. Only in a very small proportion of cases you would make an error when claiming this.

I think that the term 'by chance' is not clearly defined as long as you do not have a specific hypothesis you want to test.

You could regard the full text as your population. The complete 'census' of all words resulted in the 'true' parameter $\theta=.27$, say.

Now you describe that you took a 'sample' of words, whose characteristic is page position (first word on each page) and you want to test the hypothesis, whether page position affects the probability of a word being unique.

Hence you want to test: $$H_0:\theta=.27$$ which is equivalent to asking whether the sample of words comes from the population of all words (your full text) or forms an own (sub-) population.

If we regard the 44 pages (words) as independent draws from a Bernoulli distribution, the number of positive outcomes $X$ is Binomial. Now we need

$$P(X \ge 33|H_0) \approx 4.68*10^{-11}$$

As you can verify using R pbinom(32,44,.27,lower.tail=FALSE). This probability is very small, so you can say with very low probability of error that observing 33 unique of 44 words was not caused by chance alone, if the null hypothesis was true. Hence, $\theta$ of the sub-population of words at the top of all pages seems to be different from your population $\theta$ of .27.

Put differently, position seems to have an impact on the probability of uniqueness. Only in a very small proportion of cases you would make an error when claiming this.

I think that the term 'by chance' is not clearly defined as long as you do not have a specific hypothesis you want to test.

You could regard the full text as your population. The complete 'census' of all words resulted in the 'true' parameter $\theta=.27$, say.

Now you describe that you took a 'sample' of words, whose characteristic is page position (first word on each page) and you want to test the hypothesis, whether page position affects the propbability of a word being unique.

Hence you want to test: $$H_0:\theta=.27$$ which is equivalent to asking whether the sample of words comes from the population of all words (your full text) or forms an own (sub-) population.

If we regard the 44 pages (words) as independent draws from a Bernoulli distribution, the number of positive outcomes $X$ is Binmoial. Now we need

$$P(X \ge 33|H_0) \approx 4.68*10^{-11}$$

As you can verify using R pbinom(32,44,.27,lower.tail=FALSE). This probability is very small, so you can say with very low probability of error that observing 33 of 44 words to be unique was not observed by chance if the null hypothesis was true. Put differently, position seems to have an impact on the probablity of uniqueness. Only in a very small proportion of cases you would make an error when claiming this.

I think that the term 'by chance' is not clearly defined as long as you do not have a specific hypothesis you want to test.

You could regard the full text as your population. The complete 'census' of all words resulted in the 'true' parameter $\theta=.27$, say.

Now you describe that you took a 'sample' of words, whose characteristic is page position (first word on each page) and you want to test the hypothesis, whether page position affects the propbability of a word being unique.

Hence you want to test: $$H_0:\theta=.27$$ which is equivalent to asking whether the sample of words comes from the population of all words (your full text) or forms an own (sub-) population.

If we regard the 44 pages (words) as independent draws from a Bernoulli distribution, the number of positive outcomes $X$ is Binmoial. Now we need

$$P(X \ge 33|H_0) \approx 4.68*10^{-11}$$

As you can verify using R pbinom(32,44,.27,lower.tail=FALSE). This probability is very small, so you can say with very low probability of error that observing 33 unique of 44 words was not caused by chance alome, if the null hypothesis was true. Hence, $\theta$ of the sub-population of words at the top of all pages seems to be different from your population $\theta$ of .27. 

Put differently, position seems to have an impact on the probablity of uniqueness. Only in a very small proportion of cases you would make an error when claiming this.