Number of arms
What is the probability that the next person you meet has an above average number of arms?
Age
11 to 14
Challenge level
Problem
How likely is it that the next person you meet has an above average number of arms, where...
a) average means 'mean'?
b) average means 'mode'?
c) average means 'median'?
This problem is adapted from the World Mathematics Championships
Student Solutions
We begin by assuming that almost everybody has 2 arms. Nobody has 3, but a small number of people have lost one or both arms in accidents or amputations. Here, we will ignore fractional arms - but you don't actually have to.
| Measure | What it is | Value | More than this many arms |
|---|---|---|---|
| Mean | If all of the arms in the world were shared equally between all of the people | Just below 2 | 2 |
| Median | The middle number, when everybody's number of arms is written in a long, ordered list: 0, ..., 0, 1, ..., 1, 2, ..........................................................., 2 |
2 | 3 |
| Mode | The most common number of arms | 2 | 3 |
The next person you meet will probably have 2 arms, and definitely won't have more.
So it is almost certain that they have more than the mean number of arms, but impossible that they have more than the median or modal number of arms.