Counting the chances: sample spaces, tables and trees

Before you roll a single die, you can already work out how likely each result is. This is called theoretical probability, and it starts with a list of every possible outcome.

Sample spaces

The sample space is the list of all possible outcomes. For one die it is 1, 2, 3, 4, 5, 6. For a coin it is heads, tails. If all the outcomes are equally likely, the probability of an event is:

Tables for two things at once

When you roll two dice, there are 6 × 6 = 36 possible outcomes. A table with one die across the top and the other down the side shows them all. Fill it with the sums and you will see that 7 appears six times, but 2 and 12 appear only once each. So the chance of rolling a total of 7 is 6/36, or 1/6, while the chance of rolling 12 is only 1/36.

Tree diagrams

A tree diagram shows outcomes step by step. For two coin flips, the first flip branches into H and T, then each of those branches again. You end with four outcomes: HH, HT, TH, TT. The chance of exactly one head is 2/4 = 1/2.

Complementary events

The complement of an event is everything else: the event not happening. An event and its complement always add to 1. If the chance of rolling a six is 1/6, the chance of not rolling a six is 1 − 1/6 = 5/6.

Complements are a handy shortcut. What is the chance of getting at least one six when you roll two dice? The chance of no sixes is 5/6 × 5/6 = 25/36. So the chance of at least one six is 1 − 25/36 = 11/36, a little under a third.

Theory tells you what should happen in the long run. Experiments tell you what does happen. Good probability detectives use both.