Introduction to probability

The probability scale

Probability measures how likely something is to happen. It is a number from 0 to 1.

• A probability of 0 means the event is impossible: it can never happen.
• A probability of 1 means the event is certain: it must happen.
• A probability of 0.5 (or $\frac{1}{2}$, or 50%) is an even chance.
• Between 0 and 0.5 an event is unlikely; between 0.5 and 1 it is likely.

Why the scale stops at 0 and 1. A probability tells you what fraction of the time an event happens in the long run. Nothing can happen less often than never (0) or more often than every time (1). So a probability is never negative and never bigger than 1. If your working gives $1.3$ or $-0.2$, there is a mistake somewhere: go back and find it.

Words such as "likely" only describe a probability roughly. Numbers let you compare: an event with probability 0.7 is more likely than one with probability 0.6, even though both are "likely".

The scale as a line

Impossible events sit at 0, an even chance sits at 0.5 and certain events sit at 1. Every probability lies somewhere on this line.

Outcomes, events and equally likely outcomes

Some words you need:

• A trial is one go at an experiment: one roll of a dice, one spin of a spinner, one counter taken from a bag.
• An outcome is one possible result, such as rolling a 4.
• An event is one outcome or a group of outcomes, such as "rolling an even number" (the outcomes 2, 4 and 6).

When every outcome is equally likely,

$$\text{probability of an event}=\frac{\text{number of outcomes in the event}}{\text{total number of possible outcomes}}$$

Why this works. If there are 6 equally likely outcomes, each one gets an equal share of the certainty, $\frac{1}{6}$. An event made of 3 of them gets three shares: $\frac{3}{6}$.

Outcomes are equally likely when the question says the dice, coin or spinner is fair, or that an object is chosen at random. Be careful about *what* is equally likely. In a bag of 5 red and 15 blue counters, each counter is equally likely to be taken, but the two colours are not: red has probability $\frac{5}{20}$, not $\frac{1}{2}$.

For example, an ordinary fair dice is rolled. The probability of a number greater than 4 is $\frac{2}{6}=\frac{1}{3}$, because two of the six faces (5 and 6) are greater than 4.

On the Core papers no special notation is needed: describe each event in words, such as "the probability that the counter is red".

The probability of an event not occurring

On any trial an event either happens or it does not. Exactly one of these two must be true, so together they are certain and their probabilities add up to 1. That gives

$$\text{probability that an event does not occur}=1-\text{probability that it does occur}$$

For example, if the probability that a counter is blue is 0.8, the probability that it is not blue is $1-0.8=0.2$.

The same idea works for any list of outcomes that cannot happen together. If a spinner can land on red, green, blue or yellow, it must land on exactly one of them, so the probabilities of all the outcomes add up to 1. This lets you find a missing probability in a table: add up the ones you know and subtract from 1.

Without a calculator, take care with $1-0.07$. Write 1 as $1.00$ so the places line up: $1.00-0.07=0.93$. A quick check is to add your answer back on: $0.93+0.07=1$.

Giving a probability as a fraction, decimal or percentage

A probability can be written as a fraction, a decimal or a percentage: $\frac{3}{4}$, $0.75$ and $75\%$ are all the same probability. Use whichever the question asks for; if it does not say, any of them is fine.

Some things are not acceptable as a probability and lose the mark:

• words such as "3 out of 4" or "3 in 4";
• a ratio such as $3:4$ or $3:1$;
• the number of items, such as 3, instead of the probability.

A fraction does not have to be simplified unless the question says so, but it is good practice to simplify when it is easy. If a fraction gives a recurring decimal, such as $\frac{4}{15}=0.2666\dots$, either leave it as a fraction or give the decimal correct to 3 significant figures, $0.267$. Rounding to $0.27$ is not accurate enough.

Probabilities from tables, charts and Venn diagrams

Many questions give data about a group and then choose one member at random. Each member is then equally likely to be chosen, so

$$\text{probability}=\frac{\text{number in the group you want}}{\text{total number}}$$

The steps are always the same:

1. Find the total. In a frequency table or bar chart, add up all the frequencies.
2. Find how many are in the group the question describes. Read the wording carefully: "at least 2" includes 2; "more than 2" does not.
3. Divide.

Venn diagrams. The number in each region counts the members in that region only. For two sets $A$ and $B$:

• "in $A$" means the whole circle $A$: the part in $A$ only plus the overlap;
• "in both" means the overlap only;
• "in exactly one" means the two parts that are not shared;
• "in neither" means the region outside both circles.

The total is the sum of every region, including the outside one. A common slip is to forget the number outside the circles.