Mathematical Problem 2: Solution

The problem (set on 26 March 2010) was this:

You choose eight different books randomly from a library shelf. If you turn to the same numbered page in each book (for example, page 5), how likely is it that on at least two of these pages the text will start with the same letter?

The correct response from the options is ‘highly likely’. I tried this experimentally. With 50 trials I recorded 48 Yes results. So my estimate of the probability of at least two pages starting with the same letter is 0.96! To me that seems counter-intuitively high.

This is the best way of approaching this problem – doing lots of trials and using the relative frequency of ‘Yes’ as an estimate of probability.

You can approach it theoretically – but to make it accessible we have to assume that each of the 26 letters is equally likely. On this basis the probability of the second book producing a different letter from the first is 25/26.

Then the probability of the third book producing a different letter from the first two is 24/26. And the fourth book being different from the first three is 23/26. And so on … until we get a probability of 19/26 for the eighth book being different from the first seven looked at.

For all the books to be different you need all these things to happen, so you multiply together the probabilities: 25/26 X 24/26 X 23/26 X 22/26 X 21/26 X 20/26 X 19/26, which equals approximately 0.3. So, the probability of them not being all different is 0.7.

Why is this so different from my experimental result? This is because of the assumption that all letters are equally likely. This is clearly not the case. In my trials I did not have any examples of the first letter being x or z (I didn’t use algebra books!). And certain letters, such as t, a and s turned up much more frequently than 1 in 26. This increases substantially the probability of letters being duplicated.