Count the possibilities.

Probability compares favourable outcomes with all equally likely outcomes. The complement is useful for “at least one”: find the chance of none, then subtract from one. For successive independent trials, multiply the stage probabilities.

Without replacement, later probabilities change. A two-way table makes the sample space explicit. When a question says “given that”, restrict attention to the specified row or column before forming the fraction.

INTERACTIVE MODEL

Try it. Watch it change.

Counts in a two-way table
Event	B	not B	Total
A	8	12	20
not A	15	10	25
Total	23	22	45

A and B count8A and not B count12not A and B count15

P(A) = 0.4444 · P(B) = 0.5111 · P(A | B) = 0.3478
Not independent (or conditioning event has zero probability)

Conditioning on B restricts the denominator to the B column. Cell counts represent an illustrative population.

Explore: Change the numbers in the two-way table. Compare $P(A)$ , $P(B)$ and P(A | B).

Complement

P(\overline A)=1-P(A)

WORKED EXAMPLE

A fair coin is tossed three times. Find the probability of at least one head.

P(no heads) = ( $\frac{1}{2}$ )³ = $\frac{1}{8}$ .
P(at least one head) = 1 − $\frac{1}{8}$ = $\frac{7}{8}$ .

Assumed knowledge

Fractions, multiplication and sample spaces.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use complements and two-way tables.
Distinguish independent from dependent trials.

Year 10 preparation ↗

READY TO TRY IT?

Count the possibilities.

A fair coin is tossed three times. Find the probability of at least one head.

Put the idea to work.