The probability of an event

The probability of an event

When all the outcomes of an experiment are equally likely, the probability of an event $E$ is

where $m$ is the number of outcomes favourable to $E$ and $n$ is the total number of outcomes. Mathematicians call this the theoretical or classical probability. It is a number, usually written as a fraction in its lowest terms.

A worked example. One fair die is rolled. Find the probability of each event.

1. A number more than $4$. The favourable outcomes are $5$ and $6$: two of the six. $P = \dfrac{2}{6} = \dfrac{1}{3}$.
2. An odd number. $1, 3, 5$: three of six. $P = \dfrac{3}{6} = \dfrac{1}{2}$.
3. A $4$. One of six. $P = \dfrac{1}{6}$.

Another example. One letter is picked at random from the word BANANA. There are six letters, so six equally likely outcomes. Three of them are A, so $P(\text{A}) = \dfrac{3}{6} = \dfrac{1}{2}$. Two are N, so $P(\text{N}) = \dfrac{2}{6} = \dfrac{1}{3}$. One is B, so $P(\text{B}) = \dfrac{1}{6}$.

The formula needs equally likely outcomes. For the unequal spinner of the last lesson, red covers $180^\circ$ of $360^\circ$, so its chance is $\dfrac{180}{360} = \dfrac{1}{2}$. Counting "one colour out of three" would give the wrong answer.

From 0 to 1

The number of favourable outcomes is never less than $0$ and never more than the total. So

• An event that cannot happen is impossible. It has no favourable outcomes, so its probability is $0$. A $7$ on one die is impossible.
• An event that must happen is sure, or certain. Every outcome favours it, so its probability is $1$. A number less than $7$ on one die is sure.
• Every other event has a probability between $0$ and $1$. The closer it is to $1$, the more likely the event.

A probability can be written as a fraction, a decimal or a percentage: $\dfrac{1}{4}$, $0.25$ and $25\%$ are the same chance. A number such as $\dfrac{9}{8}$ or $-0.3$ can never be a probability.

The elementary events add up to 1

Each face of a die is an elementary event with probability $\dfrac{1}{6}$. Six of them give

This is always true: the probabilities of all the elementary events of an experiment add up to $1$. It gives a quick check, and a way to find one missing probability.

Theory and experiment

The probability says what happens in the long run, not in a few tries. Roll a die six times and you may see no $6$ at all. Roll it thousands of times and the share of $6$s settles close to $\dfrac{1}{6}$.

Practise with new numbers

Try it in Python

This program finds the probability of "a $5$ or a $6$" on one die, then rolls a die by computer more and more times. Watch the share of $5$s and $6$s.

Common mistakes

• Using the formula when outcomes are not equally likely. Win, lose or draw does not give $\dfrac{1}{3}$ each.
• Dividing by the wrong total. In BANANA there are six letters, not three different ones.
• A probability above $1$ or below $0$. It is always a slip: recheck the counts.
• Expecting the theory in a few tries. $\dfrac{1}{6}$ does not mean one $6$ in every six rolls.

Remember these

• With equally likely outcomes, $P(E) = \dfrac{\text{favourable}}{\text{total}}$.
• $0 \le P(E) \le 1$. Impossible is $0$; sure is $1$.
• The probabilities of all the elementary events add up to $1$.
• The probability tells what happens in the long run.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A spinner has eight equal sectors numbered $1$ to $8$. What is the probability of a multiple of $3$?

2. One letter is picked at random from SCHOOL. What is $P(\text{O})$?

3. Can $\dfrac{5}{4}$ be the probability of an event?

4. A spinner shows only red or white. $P(\text{red}) = 0.7$. What is $P(\text{white})$?

5. Harsh rolls a die $12$ times and gets no $6$. Is the die unfair?