Probability

The probability scale

Probability measures how likely an event is, on a scale from 0 to 1:

• 0 means impossible, 1 means certain.
• $\frac{1}{2}$ is an even chance; closer to 1 means more likely.
• Probabilities can be written as fractions, decimals or percentages — fractions are the usual exam form.

Calculating probability

When every outcome is equally likely:

$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$

Picking a letter at random from CRICKET: there are 7 letters and C appears twice, so $P(C) = \frac{2}{7}$.

A standard deck has 52 cards: 4 suits of 13, with hearts and diamonds red, clubs and spades black. So $P(\text{black}) = \frac{26}{52} = \frac{1}{2}$.

Complementary events

The complement of an event is that the event does not happen. The two probabilities always add to 1:

$$P(\text{not } A) = 1 - P(A)$$

If the probability of rain tomorrow is $0.3$, the probability of no rain is $0.7$. Watch for the words “not”, “at least” and “misses” — they usually signal that the complement is the quick route.

Listing sample spaces

The sample space is the list of every possible outcome. For one coin it is heads or tails; for one die it is 1 to 6. When two things happen together, multiply: a coin and a die give $2 \times 6 = 12$ outcomes. List them systematically (H1, H2, … T6) or use a grid so nothing is missed. Once the sample space is written out, probability is just counting.

Probability from diagrams

Venn diagrams and two-way tables are counting tools for probability. The total shown in the diagram is the denominator; the region the question asks about is the numerator.

Using the hockey and cricket Venn diagram from the sets topic: $P(\text{hockey only}) = \frac{9}{28}$. In a two-way table, complete the row and column totals first, then read off the cell you need. Translate the wording: “both” means the intersection, “at least one” means the union, “neither” means outside the circles.