The law of total probability

What you will be able to do

• split a situation into cases that do not overlap and that cover everything;
• find the probability of an event with the law of total probability;
• read the answer off a tree diagram by adding the paths that end in the event.

A faulty bolt

A factory has three machines. Machine $M_1$ makes half of the bolts, $M_2$ makes three tenths and $M_3$ makes the rest. The machines are not equally good. About $2\%$ of the bolts of $M_1$ are faulty, $3\%$ of $M_2$ and $5\%$ of $M_3$. You pick a bolt at random from the whole output. What is the chance that it is faulty?

You cannot answer with one number from the list, because the bolt comes from one machine or another. So we split the question by machine, answer it for each machine, and add the answers with the right weights. That is the law of total probability.

A partition

Events $E_1, E_2, \dots, E_n$ form a partition of the sample space when

• they are mutually exclusive: no two can happen together;
• they are exhaustive: at least one of them always happens, so their probabilities add up to $1$;
• each has positive probability.

The three machines are a partition. A bolt is made by exactly one machine. The simplest partition has two parts: an event $B$ and its complement $B'$.

The law of total probability

Let $A$ be any event. Because the $E_i$ cover everything, $A$ is the union of the pieces $A \cap E_1, A \cap E_2, \dots$, and these pieces do not overlap. So their probabilities add:

The second step uses the multiplication rule from the previous course. In words: for each case, multiply the probability of the case by the probability of $A$ in that case, then add. For two cases this is

The tree diagram shows the same idea. Each path that ends in "faulty" has the probability of its machine times the faulty rate of that machine. The answer is the sum of those paths.

A worked example with the machines

1. The paths. $M_1$ and faulty: $0.5 \times 0.02 = 0.01$. $M_2$ and faulty: $0.3 \times 0.03 = 0.009$. $M_3$ and faulty: $0.2 \times 0.05 = 0.01$.
2. The sum. $P(\text{faulty}) = 0.01 + 0.009 + 0.01 = 0.029$.
3. Check. The overall rate is $2.9\%$. It lies between the lowest rate $2\%$ and the highest rate $5\%$, and it is nearer to the rates of the big machines, which is right.

A second example with urns

A fair die is thrown. If it shows $1$ or $2$, a ball is drawn from Urn I, which holds $3$ red and $2$ white balls. Otherwise a ball is drawn from Urn II, which holds $1$ red and $4$ white balls. Find the probability that the ball is red.

The cases are "Urn I" with probability $\dfrac{1}{3}$ and "Urn II" with probability $\dfrac{2}{3}$. Then

Practise with new numbers

Try it in Python

The program adds the three paths of the factory with exact fractions. It also checks that the shares of the machines add up to $1$, which any partition must do.

Common mistakes

• Cases that do not cover everything. If the shares add up to less than $1$, a case is missing.
• Cases that overlap. A bolt cannot come from two machines. If cases overlap, one path is counted twice.
• Adding the conditional probabilities. Do not add $0.02 + 0.03 + 0.05$. Weight each one by the share of its machine first.
• Using the wrong complement. The probability of the second case in a two-case partition is $1 - P(B)$.

Remember these

• A partition is a set of events that are mutually exclusive and exhaustive, with positive probabilities.
• $P(A) = \sum_i P(E_i)\,P(A \mid E_i)$.
• In a tree, add all the paths that end in $A$.
• The answer lies between the smallest and the largest of the $P(A \mid E_i)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Three events have probabilities $0.3$, $0.45$ and $0.25$. Their probabilities add up to $1$. Is that enough for them to form a partition?

2. Given $P(B) = 0.3$, $P(A \mid B) = 0.8$ and $P(A \mid B') = 0.2$, find $P(A)$.

3. A box has $100$ bulbs. Supplier X made $60$ of them, and $5\%$ of these are faulty. Supplier Y made the other $40$, and $10\%$ of these are faulty. A bulb is chosen at random. Find the probability that it is faulty.

4. A student passes a test with probability $0.9$ if she has studied and $0.4$ if she has not. The chance that she studied is $0.7$. Find the probability that she passes.

5. Why must the cases of a partition be mutually exclusive?

References

• OpenStax, Introductory Statistics, Chapter 3, section on tree and Venn diagrams.
• NCERT, Mathematics Part II for Class XII, Chapter 13, section on partition of a sample space and the theorem of total probability.