2.5 The Law of Total Probability
2.5.1 Example: Faulty Goods in a Factory
A chocolate factory has four production lines, each contributing a different percentage to the total output and having its own unique fault rate. These are summarised in the table below:
| Line | A | B | C | D |
|---|---|---|---|---|
| % Faulty | 1 | 3 | 2.5 | 2 |
| % Output | 35 | 20 | 24 | 21 |
The management wants to understand what is the probability that a box chosen at random from the factory’s output is faulty?
To solve this, we must recognise that a faulty box can come from one of the four lines:
- line A,
- or line B,
- or line C,
- or line D.
These are mutually exclusive pathways. The total probability of a faulty box, P(F), is therefore the sum of the probabilities of a box being:
- faulty and from line A,
- faulty and from line B,
- faulty and from line C,
- faulty and from line D.
We are interested in calculating the overall probability \(P(F)\), which is just a simple sum of four intersections: \[\begin{equation*} P(F) = P(F \cap A) + P(F \cap B) + P(F \cap C) + P(F \cap D), \end{equation*}\] because the events are mutually exclusive and a faulty chocolate box can come only from one of four lines. So, we need to calculate the joint probabilities for each line, which can be done using the General Multiplication Law. For example, for line A: \[\begin{equation*} P(F \cap A) = P(F|A) \times P(A). \end{equation*}\] Similar formulae can be used for the lines B, C and D. Inserting the available probabilities, we get:
- Probability of a faulty box from Line A: \(P(F \cap A) = 0.01 \times 0.35 = 0.0035\)
- Probability of a faulty box from Line B: \(P(F \cap B) = 0.03 \times 0.20 = 0.0060\)
- Probability of a faulty box from Line C: \(P(F \cap C) = 0.025 \times 0.24 = 0.0060\)
- Probability of a faulty box from Line D: \(P(F \cap D) = 0.02 \times 0.21 = 0.0042\)
And after that, we can sum these up to get the answer: \[\begin{equation*} P(F) = 0.0035 + 0.0060 + 0.0060 + 0.0042 = 0.0197 \end{equation*}\] So, the probability of a randomly chosen box being faulty is 1.97%.
2.5.2 Formalising the Law of Total Probability
The Law of Total Probability is a law, explaining how the probability of an event can be found by considering all the different, mutually exclusive ways it can occur. This is a cornerstone of strategic thinking. When you need to find an overall probability but your data is broken down by different scenarios or categories, this law shows you how to piece it all together. Its importance lies in its ability to aggregate risk and information.
The factory example provides an illustration of the general formal rule. The Law of Total Probability states that if events \(A_1, A_2, \dots, A_n\) are mutually exclusive and collectively exhaustive (meaning they cover all possibilities), then the probability of another event B can be calculated by weighing and summing the conditional probabilities of B: \[\begin{equation} P(B) = P(B|A_1) \times P(A_1) + P(B|A_2) \times P(A_2) + \dots + P(B|A_n) \times P(A_n) \tag{2.8} \end{equation}\] This can also be interpreted as the sum of intersections of probabilities.
This law provides a fundamental bridge to more advanced concepts, most notably Bayes’ Theorem (discussed in Section 2.6), by allowing for the calculation of an overall (or marginal) probability from a set of conditional probabilities.