Complementary events

Not E

Every event has a partner: the event that it does not happen. For an event $E$, this is written $\text{not } E$, or $\overline{E}$, and it is called the complement of $E$.

• On one die, the complement of "a prime number" is "not a prime number": the outcomes $1$, $4$ and $6$.
• For a letter picked from PLANT, the complement of "a vowel" is "a consonant".
• The complement of "the bus is late" is "the bus is not late".

Every outcome favours exactly one of $E$ and $\text{not } E$. So if $E$ has $m$ favourable outcomes out of $n$, then $\text{not } E$ has the other $n - m$:

So

Using the complement

A worked example. One letter is picked at random from the word EXAMINATION. Find the probability that it is not a vowel.

1. Count. The word has $11$ letters. The vowels are E, A, I, A, I and O: $6$ of them.
2. The event. $P(\text{vowel}) = \dfrac{6}{11}$.
3. The complement. $P(\text{not a vowel}) = 1 - \dfrac{6}{11} = \dfrac{5}{11}$.
4. Check. The consonants X, M, N, T and N are $5$ letters: $\dfrac{5}{11}$ again.

When it saves work. A ticket is drawn from tickets numbered $1$ to $50$. What is the chance it is not a multiple of $9$? Counting the multiples is quick: $9, 18, 27, 36, 45$, five tickets. So $P(\text{not a multiple of } 9) = 1 - \dfrac{5}{50} = \dfrac{9}{10}$. Listing the forty-five other tickets would take much longer.

Working back. The chance that Ritu wins a game of chess against her brother is $0.45$, and the game cannot be drawn. Then the chance that she does not win, which here means her brother wins, is $1 - 0.45 = 0.55$.

Be careful in the last example. If a draw were possible, "Ritu does not win" would include draws, and it would not be the same as "her brother wins".

Practise with new numbers

Try it in Python

Python's Fraction keeps every answer exact. This program finds the chance of a vowel in a word, takes it away from $1$, and checks that the two add up to $1$.

Common mistakes

• Taking the complement of the wrong event. "Not a $6$" on a die is $1, 2, 3, 4, 5$, not just the odd numbers.
• Subtracting from the total count instead of from $1$. $P(\text{not } E) = 1 - P(E)$, not $6 - P(E)$.
• Treating "does not win" as "loses" when draws are possible. The complement includes every other result.
• Forgetting that $P(E)$ and $P(\text{not } E)$ must add to exactly $1$. Use this as a check.

Remember these

• The complement of $E$ is $\text{not } E$: every outcome that does not favour $E$.
• $P(E) + P(\text{not } E) = 1$, so $P(\text{not } E) = 1 - P(E)$.
• When the event itself is hard to count, count its complement and take it away from $1$.
• Check that the two probabilities add up to $1$.

Check yourself

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

1. $P(E) = \dfrac{3}{8}$. What is $P(\text{not } E)$?

2. On one die, what is the complement of "a number more than $2$"?

3. The chance of rain tomorrow is $0.6$. What is the chance of no rain?

4. A ticket is drawn from tickets numbered $1$ to $40$. What is the chance that it is not a multiple of $10$?

5. Can an event and its complement both have probability $0.6$?