Complements, exclusive events and the addition rule

What you will be able to do

• use the complement rule $P(\text{not } A) = 1 - P(A)$;
• tell when two events are mutually exclusive, and add their probabilities;
• use the addition rule $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ with a Venn diagram or a count.

A club survey

A made-up survey asks $120$ students two questions. Do you play a sport? Do you play an instrument? $54$ play a sport, $40$ play an instrument and $18$ do both. Pick one student at random. What is the chance that the student plays a sport or an instrument?

A quick answer adds the two groups: $54 + 40 = 94$. But the $18$ students who do both are in each group, so they were counted twice. The right count is $94 - 18 = 76$. So

The words we need

• The complement of an event $A$ is "not $A$": every outcome that is not in $A$. Books write it $A'$, $\bar{A}$ or $A^c$.
• $A \cup B$ is "$A$ or $B$ or both". It is the union.
• $A \cap B$ is "$A$ and $B$". It is the intersection, the overlap.
• Two events are mutually exclusive when they cannot happen together, so $A \cap B$ is empty.

The complement rule

An event either happens or it does not. So

In the survey, $120 - 76 = 44$ students play neither, so $P(\text{neither}) = \dfrac{44}{120} \approx 0.367$. Check: $1 - 0.633 = 0.367$. The complement is useful when "not $A$" is easier to count than $A$.

Two kinds of "or"

When $A$ and $B$ are mutually exclusive, nothing is counted twice, and

A fair die cannot show a $1$ and an even number at once. So $P(1 \text{ or even}) = \dfrac{1}{6} + \dfrac{3}{6} = \dfrac{2}{3}$.

When the events can overlap, take the overlap away once. This is the addition rule:

For the survey: $\dfrac{54}{120} + \dfrac{40}{120} - \dfrac{18}{120} = \dfrac{76}{120}$. The first rule is the second rule with $P(A \cap B) = 0$.

Try it in Python

Python has sets: collections of names in which each name appears once. The sign | joins two sets, and & keeps only what is in both. The program builds a small made-up club and checks the addition rule by counting.

Your turn

The usual mistakes

• Counting the overlap twice. Add the two events, then subtract the overlap once.
• Subtracting the overlap for exclusive events. If $A$ and $B$ cannot happen together, the overlap is $0$ already.
• Mixing up "or" and "and". "Or" is the union, and it is at least as big as each event. "And" is the overlap, and it is at most as big as each event.
• Forgetting the outside. In a Venn diagram, some people are in neither circle. They count in the total.

Remember these

• $P(\text{not } A) = 1 - P(A)$.
• Mutually exclusive events: $P(A \cup B) = P(A) + P(B)$.
• Any two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• You meet the exam form of these rules, with cards, dice and committees, in the Maths course on probability by counting.

Check yourself

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

1. The probability that a made-up train is full is $0.36$. What is the probability that it is not full?

2. A fair die is rolled. Are the events "a $2$" and "an odd number" mutually exclusive? Find the probability of "a $2$ or an odd number".

3. For two events, $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cap B) = 0.2$. Find $P(A \cup B)$ and the probability of neither.

4. In a made-up group of $40$ people, $25$ drink tea, $18$ drink coffee and $7$ drink neither. How many drink both?

5. Why can $P(A) + P(B)$ be more than $1$, while $P(A \cup B)$ never is?

References

• OpenStax, Introductory Statistics, Chapter 3, the sections on the two basic rules of probability and on Venn diagrams.