The questions at school they are always scary, even more so if preceded by the suspense of the extractions made with the book page method. This method consists ofopen a random page and add the digits of the page number until you get a class register number. But is it a right method? Actually no, it’s a method “rigged” in which some numbers have a much higher probability of being drawn than others, but be careful, this is not always the case. Let’s see where the trick lies and how we can modify this method to make it more fair.
The method of extractions from the pages of the book: “injustice” is a question of probability
The extraction method from the pages of the book is simple and works more or less like this, although there are some variations:
- one opens random page of a bookfor example the number 678, if the number obtained corresponds to a number in the list of students then this will be the number drawn, otherwise we continue with the next step, as in our case
- Yes add the numbers on the pagein our case 6+7+8=21, if the number obtained is part of the list on the register will be the number drawnotherwise add it again and check again, etc.
In our example, if there were more than 30 students in the class, then 21 would be the number drawn, but if the class was made up of only 20 students we would have to add further and extract the number 2 + 1 = 3.
But Why this method Isn’t that fair? After all, the pages are randomly drawn, so what’s the catch? It’s a question of probability, let’s try to understand it with a simplified example.
Let’s consider a book that has exactly 100 pagesnumbered from 1 to 100, and let’s imagine that all pages have the same probability of being extracted, which is difficult in reality). Now let’s see how many chance there are those who, in a class of 20 people, number 3 is drawn.
Using this method the number 3 is obtained only with pages 3, 21 and 30 since 2+1=3 and 3+0=3. Page number 12, however, is not good because 12 is part of the list of pupils and if this number comes up the question will be for pupil 12 in the register. In essence, out of 100 pages that we can open at random only 3 pages lead to the extraction of the number 3, which therefore It has a 3 in 100 chance to be extracted, that is, a 3% chance.
Now let’s see what happens with the number 12this number is drawn when the pages appear 12, 39, 48, 57, 66, 75, 84 and 93: that’s 8 pages out of 100, then the probability of drawing 12 is 8% which is more than double the chance of drawing the number 3!
If we then make the comparison with the probability of extracting the number 1 things get even worse because in that case the probability of extraction is only 2%or ¼ of the probability of extracting the number 12. Undoubtedly with this method it is better to have a name that begins with the letter A!
In short, each number has a different probability of being drawnand this makes the method not fair.
A fairer alternative variant for surprise questions
There are various fair ways to extract names for questions, but opening a book has its own charm, and some teachers use a alternative variant of the book methodwhich makes it a lot fairerand it works like this:
- one opens random page
- is calculated remainder of the division of the page number by the number of students in the classthis remainder is the extracted number, and if the remainder is 0, the last number in the list is extracted.
For example, if we have a class of 25 students and we open the book to page 54The number drawn will be 4 since 25 is in 50 twice with a remainder of 4 (54 = 25 × 2 + 4).
Let’s try to see if this method is fair by testing it with the same situation as before, a class of 20 students and a 100-page book. As before, let’s start with probability of drawing the number 3: the numbers from 1 to 100 which when divided by 20 give a remainder of 3 are the numbers 3, 23, 43, 63 and 83, we therefore have 5 chances out of 100 of extracting the number 3, one 5% chance. As for the number 12however, the numbers 12, 32, 52, 72, 92 are fine, which are again 5, therefore the 5%. The same goes for the number 1which is obtained with pages 1, 21, 41, 61 and 81, 5% chance. In short, with a 100-page book and a class of 20 students, all students have a 5% chance to be selected for the query.
Things can get slightly unbalanced if the number of pages in the book is not an exact multiple of the number of students in the class. For example, if in our case the book had 101 pages, student number 1 would have an extra chance of being extracted, that of page 101, and therefore a 5.94% probability of being extracted compared to approximately 4.95% of their classmates. However, this is a small imbalance, which is reduced if the number of pages in the book is higher, for example if there were 201 pages the two probabilities would be approximately 5.47% versus approximately 4.98%. All in all these are very small differences and we can consider this method quite fair.
