Probability explained: rules, examples and practice

Example 1: a fair dice is rolled. The probability of an even number is $\tfrac{3}{6} = \tfrac{1}{2}$, because 2, 4 and 6 are even.

Example 2: a bag has 3 red, 5 blue and 2 green counters. The probability of picking blue is $\tfrac{5}{10} = \tfrac{1}{2}$, and the probability of picking green is $\tfrac{2}{10} = \tfrac{1}{5}$.

Never write a probability as a ratio ("1 : 2") or in words ("1 in 2") in an exam: use a fraction, decimal or percentage.

A probability can never be negative or greater than 1. If your answer is, check your working.

The probabilities of all the possible outcomes add up to 1. So the probability of something not happening is:

If the probability of rain tomorrow is 0.35, the probability of no rain is $1 - 0.35 = 0.65$.

This also lets you find a missing value in a table. A spinner lands on red, blue, green or yellow:

$x = 1 - (0.2 + 0.35 + 0.15) = 1 - 0.7 = 0.3$.

To predict how many times something will happen, multiply the probability by the number of trials:

Using the spinner above, the expected number of reds in 200 spins is $0.2 \times 200 = 40$. It's an estimate: the actual number will usually be close to 40, not exactly 40.

When you can't work out a probability from equally likely outcomes, you estimate it from an experiment:

If a drawing pin lands point up 46 times in 100 throws, the relative frequency is $\tfrac{46}{100} = 0.46$. More trials give a more reliable estimate. If a coin gives heads far more than half the time over many trials, it may be biased.

• OR rule (mutually exclusive events): events that can't happen at the same time. Add the probabilities: $P(A \text{ or } B) = P(A) + P(B)$. On a dice, $P(1 \text{ or } 6) = \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{3}$.
• AND rule (independent events): one event doesn't affect the other. Multiply: $P(A \text{ and } B) = P(A) \times P(B)$. Two fair coins: $P(\text{two heads}) = \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}$.

For two or more stages, a tree diagram keeps track of every combination. For overlapping groups, use a Venn diagram.

When two things happen together, such as rolling two dice, list every outcome in a grid. Two dice give $6 \times 6 = 36$ equally likely outcomes. A total of 7 happens 6 ways (1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1), so $P(\text{total of } 7) = \tfrac{6}{36} = \tfrac{1}{6}$, the most likely total.

• Adding probabilities for AND, or multiplying for OR.
• Forgetting that the total of all outcomes is 1.
• Simplifying fractions wrongly: $\tfrac{6}{36}$ is $\tfrac{1}{6}$, not $\tfrac{1}{5}$.
• Treating relative frequency as exact: it's an estimate.

1. A letter is chosen at random from the word PROBABILITY. What is the probability it's a B?
2. The probability a bus is late is 0.12. What is the probability it isn't late?
3. A biased dice lands on 6 with probability 0.25. Estimate how many 6s in 80 rolls.
4. A fair coin is flipped and a fair dice is rolled. Find P(heads and a 5).

Answers: 1. $\tfrac{2}{11}$ (11 letters, two Bs). 2. 0.88. 3. 20. 4. $\tfrac{1}{2} \times \tfrac{1}{6} = \tfrac{1}{12}$.

More practice in our probability questions.