4 ms·

The rule of three gives an upper bound estimate that each page has a 3/200 or 1.5% chance of having a typo. For the 20,000 page book, the probability of typo pe

The rule of three gives an upper bound estimate that each page has a 3/200 or 1.5% chance of having a typo. For the 20,000 page book, the probability of typo per page is still 1.5%, but you have to roll the dice 19,800 more times, each time you turn a new page. The formula for the upper bound on probability of getting through all the remaining pages without a typo is 1 - (1-3/200)^19800 Which of course is basically 1. The lower bound is 0, the upper bound is almost 1, so the confidence interval is [0, 1). Basically, no information at all. So its much to early to conclude that you're not going to find any typos in the entire book. The best way to state your conclusions is to say "after sampling 200 pages at random, we are 95% confident that the true typo-per-page rate is in the interval 0% - 1.5% or equivalently the total number of typos in the 20,000 page book is between 0 and 300."

jdmichal 8y ago
Minor correction to your otherwise excellent final statement: "... or equivalently the total number of [pages with] typos in the 20,000 page book is between 0 and 300." We're not measuring the number of typos on a page, but only whether the page contains a typo or not. So we can't speak about the number of typos.
- PeterisP 8y ago
  The math in that article is based on the assumption that typos are scarce, because that's the situation where it makes sense that you've seen 0 events. If that probability is close to zero, the probability of two typos per page is insignificantly small and can be discarded, and the expected number of typos in the book is approximately equal to the expected number of pages with typos. So in the context of these assumptions it's not necessary to consider this difference. Just as the article itself states regarding the approximation "Since log(1-p) is approximately –p for small values of p" which relies on discarding p-squared terms as insignificantly small.
  - jdmichal 8y ago
    There's no reason to be imprecise and potentially sloppy over a two-word difference. Especially when the comment is being targeted to explain something to someone who did not have a solid grasp of the material, which is obviously necessary to make a leap such as yours. And I won't even bother with the argument that presence of one typo on a page might actually increase the probability of another typo on the same page or nearby pages. The idea that this principle is based on uniform distribution is discussed enough in other threads.

jdmichal 8y ago

Minor correction to your otherwise excellent final statement: "... or equivalently the total number of [pages with] typos in the 20,000 page book is between 0 and 300." We're not measuring the number of typos on a page, but only whether the page contains a typo or not. So we can't speak about the number of typos.

PeterisP 8y ago
The math in that article is based on the assumption that typos are scarce, because that's the situation where it makes sense that you've seen 0 events. If that probability is close to zero, the probability of two typos per page is insignificantly small and can be discarded, and the expected number of typos in the book is approximately equal to the expected number of pages with typos. So in the context of these assumptions it's not necessary to consider this difference. Just as the article itself states regarding the approximation "Since log(1-p) is approximately –p for small values of p" which relies on discarding p-squared terms as insignificantly small.
- jdmichal 8y ago
  There's no reason to be imprecise and potentially sloppy over a two-word difference. Especially when the comment is being targeted to explain something to someone who did not have a solid grasp of the material, which is obviously necessary to make a leap such as yours. And I won't even bother with the argument that presence of one typo on a page might actually increase the probability of another typo on the same page or nearby pages. The idea that this principle is based on uniform distribution is discussed enough in other threads.