Basics of Probability

A classifier that says "70% chance this flat rents within a week" is speaking the language of probability. Before trusting such a number, you need to know what it means and how probabilities combine. This month the agency listed 20 flats and recorded two facts about each:

The rule

Count favourable over total; add for OR, subtract overlap; multiply for AND.

When every flat is equally likely to be picked, a probability is a count divided by a count.

• Single event: $P(\text{fast}) = 12/20 = 0.6$.
• Complement: $P(\text{not fast}) = 1 - 0.6 = 0.4$.
• AND: $P(\text{near and fast}) = 6/20 = 0.3$, the one cell where both hold.
• OR: $P(\text{near or fast}) = 0.4 + 0.6 - 0.3 = 0.7$. The overlap is subtracted because the 6 near-and-fast flats were counted in both terms. Counting directly: 8 near plus the 6 far-and-fast flats makes 14 out of 20.
• Conditional: $P(\text{fast} \mid \text{near})$ means "among near flats only". That is $6/8 = 0.75$.
• Multiplication rule: $P(A \text{ and } B) = P(A) \times P(B \mid A)$. Check: $0.4 \times 0.75 = 0.3$.

Two events are independent when knowing one does not change the other: $P(B \mid A) = P(B)$. Here $P(\text{fast} \mid \text{near}) = 0.75$ but $P(\text{fast}) = 0.6$, so metro distance tells you something. That is exactly what makes it a useful feature. Only for independent events can you multiply the plain probabilities: two unrelated flats from different months, each with a 0.6 chance, both rent fast with probability $0.6 \times 0.6 = 0.36$.

The shortcut

Shrink the room. For any "given" question, cross out every row or column that does not match the condition, then count favourable over what is left. The new total is the size of the condition, not 20.

The trap is treating $P(A \mid B)$ and $P(B \mid A)$ as the same thing. They share a numerator but have different rooms.

Drills

Answer fast — trust the rule.

Remember

• Count favourable over total; add for OR, subtract overlap; multiply for AND.
• Shortcut: for "given", shrink the room to the condition, then count.
• Trap: swapping the condition, or multiplying probabilities of dependent events.