Introduction to Probability

The probability scale from 0 to 1

Probability is a measure of how likely an event is, and the scale runs from 0 to 1. A probability of 0 means the event cannot happen at all; a probability of 1 means it is certain to happen. Everything else sits in between, with $\frac{1}{2}$ - that is 0.5, or 50% - marking an even chance.

A probability is never negative and never bigger than 1. If you work out an answer such as 1.4 or $-0.2$, you have divided the wrong way round, and the check costs you two seconds.

Cambridge accepts the answer as a fraction, a decimal or a percentage, and a mark scheme usually accepts any of the three. $\frac{3}{4}$, 0.75 and 75% are the same probability written three ways. Write the form the question asks for, and where it asks for nothing in particular, a fraction in its simplest form is never wrong.

Equally likely outcomes: favourable over total

When every outcome is equally likely, the probability of an event is the number of outcomes that count as the event, divided by the total number of possible outcomes. Roll an ordinary die and there are six outcomes, all equally likely. The probability of rolling a 4 is $\frac{1}{6}$, because one outcome out of the six counts.

'Favourable' means the outcomes the question asks about, not outcomes that are good. If a bag holds 7 red counters and 3 blue ones, the probability of red is $\frac{7}{10}$: the 7 counts, and the 10 on the bottom is the total number of counters in the bag.

The method needs the outcomes to be equally likely, and that has to be true for a reason - because the die is fair, or because the spinner has equal sectors, or because each counter is equally likely to be picked. A biased die still has six possible outcomes, but they are not equally likely, so counting them tells you nothing at all.

• Fair six-sided die: the probability of a six is $\frac{1}{6}$, and the probability of an odd score is $\frac{3}{6}=\frac{1}{2}$
• Bag of 7 red and 3 blue counters: the probability of red is $\frac{7}{10}$ = 0.7 = 70%
• Standard pack of 52 cards: the probability of a heart is $\frac{13}{52}=\frac{1}{4}$

The complement: the probability of something not happening

For any event, either it happens or it does not, so those two probabilities must add up to 1. That gives the complement rule: the probability of an event not occurring is 1 minus the probability of it occurring.

If the probability that a counter is blue is 0.8, the probability that it is not blue is $1-0.8=0.2$. If the probability of a six is $\frac{1}{6}$, the probability of not a six is $1-\frac{1}{6}=\frac{5}{6}$. In percentages, $1-75\%=25\%$. The answer always comes out as the same kind of number you started with.

The complement is the fastest route through any question containing 'at least one'. The opposite of 'at least one blue' is 'no blue at all', and working out the probability of none is usually far easier than counting the cases with one, two or three blues.

Extended candidates write the complement as $P(A^{\prime })$, read as the probability of not A, alongside $P(A)$ for the probability of A. Core candidates do not need the notation, but the subtraction is identical.

• $P(\text{blue})=0.8$, so the probability of not blue is $1-0.8=0.2$
• $P(\text{six})=\frac{1}{6}$, so the probability of not a six is $\frac{5}{6}$
• The complement of 'at least one blue' is 'no blue at all'

Listing outcomes and sample space diagrams

A sample space is the list of everything that can happen. For one die it is 1, 2, 3, 4, 5, 6. For two dice a sample space diagram lays the outcomes out in a table, which gives 36 equally likely outcomes - not 11 totals. The totals from 2 to 12 are not equally likely, however tempting they look as a list.

Work systematically: put the score on the second die down the side and the score on the first die across the top, then fill in the cells row by row. Systematic listing is worth the thirty seconds even when you think you can see the answer, because the method mark counts the outcomes and one omitted cell loses it.

Once the diagram is drawn, counting is all that is left. A total of 7 appears in 6 of the 36 cells, so its probability is $\frac{6}{36}=\frac{1}{6}$. A total of 12 appears once, in the cell where both dice show 6, so $P(12)=\frac{1}{36}$ and the probability of not 12 is $\frac{35}{36}$.

Combined events: two things happening

A combined event is two or more things happening, such as two coins, two counters, or a counter and a die. When the outcomes are equally likely and the events do not affect each other, multiply: the probability of A and B is $P(A)\times P(B)$. Two tosses of a fair coin give heads followed by heads with probability $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$.

Where a counter is taken, replaced and a second one taken, the bag is back to how it started, so the second fraction is identical to the first. From a bag of 3 red and 2 blue counters, the probability that both draws are red is $\frac{3}{5}\times \frac{3}{5}=\frac{9}{25}$. The Extended paper also sets these questions without replacement, where the second fraction changes: $\frac{3}{5}\times \frac{2}{4}=\frac{3}{10}$.

Questions with 'at least one' in them are complement questions again. The probability of at least one blue in the two draws above is $1-\frac{9}{25}=\frac{16}{25}$, because the only alternative to at least one blue is no blue at all.

A tree diagram draws the whole thing out: outcomes at the end of the branches, probabilities beside the branches, and multiply along any path to get the probability of that combination. The probabilities of all the paths add up to 1, which is the check on your arithmetic.

Probabilities from tables, graphs and Venn diagrams

Not every question hands you the counts in a sentence. A two-way table gives them row by row, and the number you need on the bottom of the fraction is the total for the group the question is about. Read the row totals, the column totals and the grand total before you write anything.

A Venn diagram does the same job with two sets. The number in both sets sits in the overlap, and the number outside both circles still belongs to the universal set, so it belongs in the denominator as well. With 30 students, 12 playing football, 10 playing hockey and 8 playing neither, the probability that a student chosen at random plays football is $\frac{12}{30}=\frac{2}{5}$, not $\frac{12}{22}$.

Read the last line of the question carefully. 'A student chosen at random from those who play hockey' changes the denominator to 10, and the same table then gives a completely different answer.

Choosing the form of your answer

A probability is a number, and the same number can be written as a fraction, a decimal or a percentage. $\frac{3}{8}$ = 0.375 = 37.5%. The marks are in the value, not the notation, but a fraction should be in its simplest form, so $\frac{6}{36}$ is written $\frac{1}{6}$ and $\frac{3}{6}$ is written $\frac{1}{2}$.

Where a question asks for a decimal, divide exactly when you can. $\frac{1}{8}=0.125$ is exact, while $\frac{1}{6}=0.1666\dots$ is not, so write 0.167 to three decimal places if the question asks for a rounded answer - and keep the fraction if it does not.

Two checks finish the job. Every probability lies between 0 and 1, and the probabilities of all the possible outcomes add up to 1. If a spinner's three colours have probabilities 0.3, 0.45 and 0.15, the sum is 0.9, so something is missing - and saying so is worth a mark when the question asks whether the figures can be right.

• $\frac{3}{8}=0.375=37.5\%$
• $\frac{6}{36}$ simplifies to $\frac{1}{6}$
• $0.3+0.45+0.15=0.9$, so those three probabilities cannot be the whole story