Probability Basics

Will it rain on sports day? Is this the winning raffle ticket? Will the next sweet you grab from the bag be a red one? All day long you weigh up how likely things are — and probability is simply the maths that turns those hunches into numbers you can compare.

Probability measures how likely something is to happen. Some things are sure to happen, some can never happen, and most are somewhere in between. Probability turns that feeling of "how likely?" into a number, so we can compare chances exactly instead of arguing about words.

Every probability is a number from 0 to 1:

0 means the event is impossible — it never happens;
1 means it is certain — it always happens;
\tfrac{1}{2} means an even chance — just as likely to happen as not.

The same probability can be written as a fraction, a decimal or a percentage — for example \tfrac{1}{2} = 0.5 = 50\%. We describe events in everyday words too, sliding from one end of the scale to the other:

\text{impossible} \;\to\; \text{unlikely} \;\to\; \text{even chance} \;\to\; \text{likely} \;\to\; \text{certain}

The sun rising tomorrow is certain (probability 1). A normal six-sided die showing a 7 is impossible (probability 0). A tossed coin landing heads is an even chance (probability \tfrac{1}{2}).

See the scale

Step through the line that every probability lives on, from impossible at one end to certain at the other.

Counting the chances

How do we find the exact number? When every result is equally likely — a fair coin, a fair die, a ball drawn from a shaken bag without peeking — we just count:

P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}

A "favourable" outcome is simply one that counts as the event happening. Three quick examples:

A coin. There are 2 equally likely outcomes — heads or tails — and 1 of them is heads, so P(\text{heads}) = \tfrac{1}{2}.
A die. A fair die has 6 faces. Rolling a 4 is just 1 of them, so P(4) = \tfrac{1}{6}. Rolling an even number (2, 4 or 6) gives 3 favourable faces, so P(\text{even}) = \tfrac{3}{6} = \tfrac{1}{2}.
Coloured balls. A bag holds 2 red balls and 3 blue balls — 5 in all. Drawing red has 2 favourable outcomes out of 5, so P(\text{red}) = \tfrac{2}{5} = 0.4.

a coloured ball Picture a bag you cannot see into. You give it a good shake and pull out one ball. Because every ball is equally likely to be the one your hand grabs, counting works perfectly: the chance of red is just the number of red balls over the total. Add more red balls and red gets likelier; the fraction — and the marker on the probability scale — slides toward 1.

a coin We keep saying a "fair" coin. Fair means every outcome is equally likely — heads and tails each get exactly \tfrac{1}{2}. A bent or weighted coin might land heads 0.7 of the time. Then the simple count rule no longer applies, because the two outcomes are not equally likely. Whenever you use favourable-over-total, check that the outcomes are genuinely fair first.

See it: drawing from a bag

Here is a bag of red and blue balls. Count the red balls, count them all, and the probability of pulling out red is just one over the other. Press Refresh for a new mixture and work out the new chance.

Three traps to dodge with probability:

A probability is always between 0 and 1 (0 \le P \le 1). If your answer comes out as \tfrac{7}{6} or 1.3, you have made a mistake — nothing can be more than certain.
The count rule P = \tfrac{\text{favourable}}{\text{total}} only works when the outcomes are equally likely — a fair die, a fair coin, a fair draw. A loaded die breaks it.
A "fair" coin or die just means every outcome is equally likely. It does not promise heads and tails will alternate — luck still happens.

Two worked examples

Spinner. A spinner is split into 4 equal sections coloured red, red, blue and green. Two of the four sections are red, so P(\text{red}) = \frac{2}{4} = \frac{1}{2} = 0.5. It is an even chance — halfway along the scale.

Not red. What is the chance the spinner does not land on red? Of the four sections, two are not red (blue and green), so P(\text{not red}) = \tfrac{2}{4} = 0.5. Notice that P(\text{red}) + P(\text{not red}) = 0.5 + 0.5 = 1. The chance of an event and the chance of it not happening always add up to 1.