Chapter 11

Probability and statistics

Probability puts a number on how likely something is, between zero for impossible and one for certain. It is counting: the outcomes you want, over the outcomes there are.

Statistics goes the other way. Given a pile of numbers, it produces one number that stands for the pile, and which one you pick changes what it tells you.

Probability and averages

Probability measures how likely something is. Statistics describes a set of numbers with a single number that stands for the group.

  • Probability is the number of outcomes you want divided by the number of outcomes there are.
  • A probability is always between 0 and 1. Zero means it cannot happen, one means it always does.
  • The mean is the total divided by how many numbers there are.
  • The median is the middle number once they are in order.
  • The mode is the number that appears most often.
  • The range is the largest number minus the smallest.

Rolling a 4 on a six-sided die

One face out of six is a 4.

1/6, which is about 0.17, or 17%.

Formulas

A few of these are usually memorised. The rest follow from what they describe: profit is what is left after the cost comes off, which is the formula itself stated in words.

Probability and averages

Probability

P = outcomes you want ÷ outcomes there are

Count both, then divide. The answer is a fraction, and it can be written as a decimal or a percent instead.

Example

A bag holds 3 red marbles and 5 blue ones.

3 ÷ 8 = 0.375

The chance of drawing red is 3/8, or 37.5%.

Mean

mean = total ÷ how many

Add every number, then share the total out equally between them.

Example

The numbers 4, 8, 6, 2.

4 + 8 + 6 + 2 = 20

20 ÷ 4 = 5

Range

range = largest − smallest

How far apart the two ends are, which says how spread out the set is.

Example

The numbers 4, 8, 6, 2.

8 − 2 = 6

Examples

Each one is set out the way a solution is written on paper: the question, the steps in order, then the answer.

Example 1. Median of an odd-sized set

Find the median of 7, 3, 9, 1, 5.

Solution

  1. Put them in order: 1, 3, 5, 7, 9.
  2. There are five numbers, so the middle one is the third.

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Example 2. Median of an even-sized set

Find the median of 2, 4, 6, 10.

Solution

  1. They are already in order.
  2. There are four numbers, so there is no single middle one.
  3. Take the mean of the middle two: (4 + 6) ÷ 2 = 5.

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