TMUA Counting and Probability

Counting

Fill positions one at a time, handling the most restricted position first. Without repetition, each choice reduces the options for the next.

Probability rules

• Mutually exclusive: $P(A\text{ or }B)=P(A)+P(B)$.
• Independent: $P(A\text{ and }B)=P(A)P(B)$.
• Conditional: $P(A\mid B)=\dfrac{P(A\text{ and }B)}{P(B)}$.

Without replacement

After each draw, the total and the number of that type both change. Draw a tree and multiply along branches.

Strategy

Decide first whether order matters. Arrangements use products, as in $5\times4\times3$, while selections where order is irrelevant use $\binom nr$.

When a question says 'at least one', it is often quicker to find the probability of the opposite event and subtract it from 1. Then check that your answer lies between 0 and 1.

Worked examples

Common mistakes

• Using the full sample space when the question gives a condition.
• Keeping the same denominator for the second draw when there is no replacement.
• Adding probabilities for 'and' or multiplying them for 'or'.
• Double-counting outcomes that fit two categories.

Test yourself on this topic with our free practice questions.