Chapter 12: Probability

The chapter problem: when the alarm sounds, is there a fire?

A warehouse has a fire-detection system. Tests give the manufacturer two reliability figures:

• when there is a fire, the alarm sounds with probability $0.95$;
• when there is no fire, the alarm still sounds with probability $0.02$.

The second figure is the false-alarm rate. A detector that cries wolf is annoying, but a detector that stays silent in a fire is dangerous.

Fires are rare. Records for this warehouse suggest that on any morning there is a real fire with probability $0.001$. That is one morning in a thousand.

Early one morning the alarm sounds. The site manager must decide whether to call the fire service. The manager knows the two reliability figures, and now needs one more number:

This is the probability of a fire, given that the alarm has already sounded. By the end of the chapter you will work it out. The answer surprises most people, and the reason for the surprise is the heart of this chapter.

First contact — no working needed yet. Before you calculate anything, predict the value of $P(\text{fire} \mid \text{alarm})$. Choose one:

• above $0.5$;
• between $0.1$ and $0.5$;
• below $0.1$.

Write your choice down, with one sentence that gives your reason. Keep the prediction for the Review.

By the end of this chapter you will be able to

• use the addition rule for mutually exclusive events;
• use the general addition rule $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$;
• test whether two events are independent, using $P(A \text{ and } B) = P(A)P(B)$;
• explain the difference between mutually exclusive events and independent events;
• find a conditional probability from a two-way table, a Venn diagram or a tree diagram;
• use the formula $P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}$ and its rearrangement $P(A \text{ and } B) = P(A \mid B)P(B)$;
• find the probability of an event by adding the tree-diagram paths that lead to it;
• work backwards from a result to its cause, as the site manager must;
• build a probability model from a real situation, state its assumptions, and criticise them;
• refine a model when an assumption fails, and describe how the answer moves.

Where it is used

Probability is how we reason when we cannot see the whole picture. A doctor reads a screening test and must judge how much a positive result means. A bank checks a payment against a model of fraud. A weather centre attaches a percentage to rain. An insurance company sets a premium from the chance of a claim. A manufacturer samples a batch instead of testing all of it.

In each case someone builds a model, states assumptions, and reads a number. The reading is only as good as the assumptions, which is why this chapter ends with criticism, not only calculation. The same ideas return in Chapter 13, where a distribution is the model, and in Chapter 14, where a test turns a probability into a decision.

Your turn to connect it

Think of a test, an alarm or a check you have met that can be wrong. It could be a smoke alarm, a phone that unlocks with your face, a security warning on a website, or a screening test. Write down what a positive or alarmed result would mean to you. Then write down one reason the result might be wrong. Keep your answer for the Review.

Roadmap

1. 12.1 Mutually exclusive and independent events
2. 12.2 Conditional probability and the conditional probability formula
3. 12.3 Modelling with probability and critiquing assumptions
4. Review