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The complement of an event

Probability, grade 9 Lesson 3 of 11Learning

The complement of an event

The complement

The complement of A\displaystyle A is the event “A\displaystyle A does not happen”: all the outcomes that are not in A\displaystyle A. It is written A‾\displaystyle \overline{A} – the letter A\displaystyle A with a bar over it.

  • When a die is thrown, A\displaystyle A – “a 6 comes up”: {6}\displaystyle \{6\}.
  • A‾\displaystyle \overline{A} – “a 6 does not come up”: {1,2,3,4,5}\displaystyle \{1,2,3,4,5\}.
  • Every outcome is in exactly one of them: in A\displaystyle A or in A‾\displaystyle \overline{A}.
The probability of the complement

The probability of an event and the probability of its complement add up to 1:

P(A‾)\displaystyle P(\overline{A}) =\displaystyle = 1−P(A)\displaystyle 1-P(A)
  • Why: the outcomes of A\displaystyle A and of A‾\displaystyle \overline{A} together are all the outcomes, and the probability of all the outcomes together is 1.
  • With the die: P(A)=16\displaystyle {P(A)}={\frac{1}{6}}, and P(A‾)=56=1−16\displaystyle {P(\overline{A})}={\frac{5}{6}}={1-\frac{1}{6}}.
Calculating through the complement

The example below calculates the probability of an event with many outcomes through its complement, which has few.

A box holds 50 cards, numbered 1 to 50.
One card is drawn without looking.
The event A\displaystyle A: the number is not divisible by 7.

  • A\displaystyle A has many outcomes, and they are hard to count.
  • A‾\displaystyle \overline{A} – “the number is divisible by 7” – is only seven outcomes: {7,14,21,28,35,42,49}\displaystyle \{7,14,21,28,35,42,49\}.
  • So P(A‾)=750\displaystyle {P(\overline{A})}={\frac{7}{50}}.
P(A)=1−750=4350 P(A)=1-\frac{7}{50}=\frac{43}{50}
Example: a given probability

The example below calculates the probability of the complement when only the event’s probability is given, with no outcomes to count.

The event A\displaystyle A: it rains tomorrow.
P(A)=0.3\displaystyle {P(A)}={0.3}.

The complement, A‾\displaystyle \overline{A}: it does not rain tomorrow.

P(A‾)=1−0.3=0.7 P(\overline{A})=1-0.3=0.7
Exercise 1

A die is thrown.
For each event, write its complement – in words and as outcomes.

a

An even number comes up.

b

A number greater than 2 comes up.

c

A 1 or a 6 comes up.

Exercise 2

A box holds 40 cards, numbered 1 to 40.
One card is drawn without looking.
Calculate the probability of each event through its complement.
Write each probability as a fraction in its lowest terms.

a

The number is not divisible by 10.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    The number is not divisible by 5.

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Exercise 3

      Calculate the probability of the complement.

      a

      The probability that the bus arrives on time is 0.85\displaystyle 0.85.
      What is the probability that it does not arrive on time?

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b

        The probability of drawing a blue marble from a bag is 27\displaystyle \frac{2}{7}.
        What is the probability of drawing a marble that is not blue?

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 4

          A bag holds 20 marbles: red, blue and green.
          One marble is drawn without looking.
          The probability of drawing a red marble is 14\displaystyle \frac{1}{4}.
          The probability of drawing a blue marble is 25\displaystyle \frac{2}{5}.

          a

          How many red marbles are in the bag? How many blue ones?

          Red

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            Blue

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b

              What is the probability that the marble is not green?

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                c

                What is the probability that the marble is green?

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  Exercise 5

                  A box holds 10 cards, numbered 1 to 10.
                  One card is drawn without looking.
                  The event A\displaystyle A: the number is less than 7.
                  The event B\displaystyle B: the number is less than 5.
                  Calculate P(A)\displaystyle P(A) and P(B)\displaystyle P(B). Is B\displaystyle B the complement of A\displaystyle A? Explain.

                  P(A)\displaystyle P(A)

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    P(B)\displaystyle P(B)

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power