Challenge Level

*This resource is part of the collection Probability and Evidence.*

This problem considers a surprising result that can have some real life implications. Read the text below, and then take a look at the questions that follow.

DNA profiling is an invaluable tool for the police and the courts to identify criminals. This is because it's extremely accurate: the chance that a random person will match a DNA profile taken from a crime scene is estimated to be less than 1 in a billion.

However, when it comes to chance and probability, things aren't always as straightforward as they seem. You have to very careful to do your sums properly before jumping to conclusions.

As an example, let's take a controversy sparked by the discovery of a lab worker in the US state of Arizona in 2001. While she was sifting through a database containing DNA profiles of 65,000 convicted criminals the lab worker found two matching profiles. The two convicts to whom the profiles belonged were unrelated. Subsequent research showed that there were many more matching pairs in the database.

This surprised people, because the chance of one profile matching a given profile is so small. Some people suggested that something must be seriously wrong with the technology used to analyse and compare DNA samples. This would pose a terrible problem because if the technology gives rise to unnaturally many matches, then many innocent people might be wrongly convicted of crimes.

However, when it comes to chance and probability, things aren't always as straightforward as they seem. You have to very careful to do your sums properly before jumping to conclusions.

As an example, let's take a controversy sparked by the discovery of a lab worker in the US state of Arizona in 2001. While she was sifting through a database containing DNA profiles of 65,000 convicted criminals the lab worker found two matching profiles. The two convicts to whom the profiles belonged were unrelated. Subsequent research showed that there were many more matching pairs in the database.

This surprised people, because the chance of one profile matching a given profile is so small. Some people suggested that something must be seriously wrong with the technology used to analyse and compare DNA samples. This would pose a terrible problem because if the technology gives rise to unnaturally many matches, then many innocent people might be wrongly convicted of crimes.

We can model a similar situation where the probability of any two random people matching is 1 in 225, and we have a database of 30 people. Each person is given a random number between 1 and 225; this represents their DNA profile.

Imgine you are one of the group of 30 people on the database.

Consider the other people in order. What is the probability that the first person has a

What is the probability that the second person has a different number to you?

What is the probability that

What is the probability that the first three people all have different numbers to you?

What is the probability that all 29 other people have different numbers to you?

What is the probability that at least one other person has the same number as you?

Does this help to explain why so many pairs were found in the Arizona database?