Mathematical Problem 2: Page numbers probability

Here's a variant of a well-known counter-intuitive question about probability.

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? (Exclude dictionaries!)

Choose one of the following:

• highly likely (probability about 0.9)
• fairly likely (probability about 0.65)
• about evens (probability about 0.5)
• fairly unlikely (probability about 0.35)
• highly unlikely (probability about 0.1)

This is a good problem to use with primary school children because it is easy for them to gather data to enable them to estimate the probability using relative frequency. Each of eight children has one of the books. They all turn to page 5 and call out the first letter on the page. They just record 'Yes' or 'No', depending on whether or not there is a letter that turns up twice. They then do the same for page 6, page 7, and so on. They can quickly gather data for a sample of, say, 50 pages and calculate the relative probability of 'Yes'.

Try it! The result may surprise you.