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Probabilities Sum to One

CONCEPTUAL
Mathematics›Probability|Ages 10—11|ID: mt_3fwYu7imd4

Understand that when all possible outcomes of a trial are listed, their probabilities must add up to 1; use this to find the probability of an event NOT happening: P(not A) = 1 − P(A); apply this shortcut to avoid counting all unfavourable outcomes directly

Mastery Evidence

  • List all outcomes of spinning a 4-colour spinner and verify their probabilities add up to 1
  • Calculate P(not rolling a 3) as 1 − 1/6 = 5/6 using the complement rule
  • Spot an error in a probability table where the values don't sum to 1 and explain what's wrong

Assessment Prompt

“If the probability of it raining tomorrow is 0.3, can [child] work out the probability of it NOT raining — and explain why all probabilities in a situation must add up to 1?”

Prerequisites2

Show full prerequisite tree
  • The 0-to-1 Probability Scale hard

    The complement rule P(not A) = 1 − P(A) requires understanding probability as a number that lies between 0 and 1

    • Probability as a Fraction hard

      The formal 0-1 probability scale formalises the fractional representation of equally-likely outcomes introduced at age 9-10

      • Simple Chance Experiments soft

        Practical experiment experience provides the intuitive grounding that makes fractional probability representation meaningful

        • Pictograms and tally charts soft

          Recording probability experiment results in tally charts uses the data-recording skills taught in Data & Statistics

          • Pictograms and tally charts (age 6+) hard

            Constructing pictograms, tally charts, and bar charts requires these display vocabulary terms

          • Sorting into categories hard

            Constructing pictograms and tally charts requires classifying and counting objects first

            • Comparing groups: more or fewer soft

              Sorting categories by count benefits from ability to compare quantities

              • Counting objects to 20 soft

                Counting a set helps when comparing groups, but younger children (GB age 4) can compare using matching without formal counting to 20

                • How Many in Total? hard

                  Answering 'how many?' requires the cardinality principle

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

                • One-to-one counting hard

                  Counting objects to answer 'how many?' requires one-to-one correspondence

            • Counting objects to 20 hard

              Counting objects in each category requires being able to count sets of objects

              • How Many in Total? hard

                Answering 'how many?' requires the cardinality principle

                • One-to-one counting hard

                  Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

              • One-to-one counting hard

                Counting objects to answer 'how many?' requires one-to-one correspondence

          • Sorting Data into Categories soft

            Data representation formats (pictograms, tally charts) support organising data

            • How Many in Total? soft

              Counting data in categories requires understanding cardinality

              • One-to-one counting hard

                Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

            • Pictograms and tally charts (age 6+) hard

              Organising and representing data requires data, tally, frequency, and category vocabulary

            • Sorting into categories hard

              Organising data in categories builds on classifying and counting objects in categories

              • Comparing groups: more or fewer soft

                Sorting categories by count benefits from ability to compare quantities

                • Counting objects to 20 soft

                  Counting a set helps when comparing groups, but younger children (GB age 4) can compare using matching without formal counting to 20

                  • How Many in Total? hard

                    Answering 'how many?' requires the cardinality principle

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

                  • One-to-one counting hard

                    Counting objects to answer 'how many?' requires one-to-one correspondence

              • Counting objects to 20 hard

                Counting objects in each category requires being able to count sets of objects

                • How Many in Total? hard

                  Answering 'how many?' requires the cardinality principle

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

                • One-to-one counting hard

                  Counting objects to answer 'how many?' requires one-to-one correspondence

        • Likelihood Language hard

          Conducting probability experiments and describing results requires knowing the language used to describe likelihood

      • Unit fractions hard

        Expressing probabilities as unit fractions (1/6, 1/2, 1/3) requires prior knowledge of unit fractions from the Fractions domain

        • Fractions of amounts hard

          Finding fractions of discrete sets extends finding fractions of shapes/quantities

          • Finding halves and quarters (age 5+) hard

            Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters

            • What Is a Half? hard

              Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4

              • Division as equal sharing hard

                Finding a half requires equal sharing into 2 groups — a division concept

                • Subtraction as taking away or separating hard

                  Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                  • How Many in Total? hard

                    Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

          • What Is a Half? hard

            Working with fractions extends from Y1 understanding of halves

            • Division as equal sharing hard

              Finding a half requires equal sharing into 2 groups — a division concept

              • Subtraction as taking away or separating hard

                Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                • How Many in Total? hard

                  Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

          • Division as equal sharing soft

            Finding fractions of quantities uses equal sharing (division)

            • Subtraction as taking away or separating hard

              Division as equal sharing/grouping requires understanding subtraction as taking away/separating

              • How Many in Total? hard

                Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                • One-to-one counting hard

                  Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

          • Fraction Notation hard

            Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator

        • Division as equal sharing soft

          Finding 1/4 of 12 objects connects to division as sharing equally

          • Subtraction as taking away or separating hard

            Division as equal sharing/grouping requires understanding subtraction as taking away/separating

            • How Many in Total? hard

              Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

              • One-to-one counting hard

                Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

      • Equally Likely Outcomes hard

        Using fractions to represent probability only makes sense for equally-likely outcomes, so the equally-likely concept must come first

        • Ordering Likelihoods hard

          Understanding what 'equally likely' means is a specific case of comparing likelihoods that requires the general comparison skill first

          • Likelihood Language hard

            Comparing and ordering likelihoods requires first knowing the vocabulary of likelihood (certain, likely, unlikely, impossible)

    • Comparing fractions soft

      Expressing probability as fractions, decimals, and percentages requires comparing and ordering fractions — a skill built in the Fractions domain

      • Decomposing a shape into more equal shares soft

        More shares = smaller helps understand why 1/5 < 1/3

        • Halves & Quarters of Shapes hard

          Comparing share sizes requires experience partitioning into halves and quarters

          • Finding halves and quarters (age 5+) hard

            Partitioning into fourths/quarters extends from Y1 understanding of quarters

            • What Is a Half? hard

              Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4

              • Division as equal sharing hard

                Finding a half requires equal sharing into 2 groups — a division concept

                • Subtraction as taking away or separating hard

                  Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                  • How Many in Total? hard

                    Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

          • What Is a Half? hard

            Partitioning shapes into halves extends from Y1 understanding of halves

            • Division as equal sharing hard

              Finding a half requires equal sharing into 2 groups — a division concept

              • Subtraction as taking away or separating hard

                Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                • How Many in Total? hard

                  Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

      • Fractions on a number line hard

        Comparing fractions requires understanding them as numbers on a line

        • Fractions of amounts hard

          Placing fractions on number line requires knowing what fractions are

          • Finding halves and quarters (age 5+) hard

            Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters

            • What Is a Half? hard

              Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4

              • Division as equal sharing hard

                Finding a half requires equal sharing into 2 groups — a division concept

                • Subtraction as taking away or separating hard

                  Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                  • How Many in Total? hard

                    Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

          • What Is a Half? hard

            Working with fractions extends from Y1 understanding of halves

            • Division as equal sharing hard

              Finding a half requires equal sharing into 2 groups — a division concept

              • Subtraction as taking away or separating hard

                Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                • How Many in Total? hard

                  Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

          • Division as equal sharing soft

            Finding fractions of quantities uses equal sharing (division)

            • Subtraction as taking away or separating hard

              Division as equal sharing/grouping requires understanding subtraction as taking away/separating

              • How Many in Total? hard

                Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                • One-to-one counting hard

                  Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

          • Fraction Notation hard

            Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator

        • Tenths soft

          Counting in tenths supports placing fractions on a number line

          • Fractions of amounts hard

            Tenths extend fraction understanding from halves, thirds, quarters

            • Finding halves and quarters (age 5+) hard

              Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters

              • What Is a Half? hard

                Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4

                • Division as equal sharing hard

                  Finding a half requires equal sharing into 2 groups — a division concept

                  • Subtraction as taking away or separating hard

                    Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                    • How Many in Total? hard

                      Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                      • One-to-one counting hard

                        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

            • What Is a Half? hard

              Working with fractions extends from Y1 understanding of halves

              • Division as equal sharing hard

                Finding a half requires equal sharing into 2 groups — a division concept

                • Subtraction as taking away or separating hard

                  Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                  • How Many in Total? hard

                    Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

            • Division as equal sharing soft

              Finding fractions of quantities uses equal sharing (division)

              • Subtraction as taking away or separating hard

                Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                • How Many in Total? hard

                  Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

            • Fraction Notation hard

              Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator

          • Counting in 2s soft

            Skip counting supports counting in tenths

  • Calculating Simple Probability soft

    Using the complement rule is easier once students can calculate basic probabilities and see that favourable + unfavourable outcomes cover all possibilities

    • The 0-to-1 Probability Scale hard

      Calculating probability using favourable/total outcomes requires understanding probability as a number on a 0-1 scale

      • Probability as a Fraction hard

        The formal 0-1 probability scale formalises the fractional representation of equally-likely outcomes introduced at age 9-10

        • Simple Chance Experiments soft

          Practical experiment experience provides the intuitive grounding that makes fractional probability representation meaningful

          • Pictograms and tally charts soft

            Recording probability experiment results in tally charts uses the data-recording skills taught in Data & Statistics

            • Pictograms and tally charts (age 6+) hard

              Constructing pictograms, tally charts, and bar charts requires these display vocabulary terms

            • Sorting into categories hard

              Constructing pictograms and tally charts requires classifying and counting objects first

              • Comparing groups: more or fewer soft

                Sorting categories by count benefits from ability to compare quantities

                • Counting objects to 20 soft

                  Counting a set helps when comparing groups, but younger children (GB age 4) can compare using matching without formal counting to 20

                  • How Many in Total? hard

                    Answering 'how many?' requires the cardinality principle

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

                  • One-to-one counting hard

                    Counting objects to answer 'how many?' requires one-to-one correspondence

              • Counting objects to 20 hard

                Counting objects in each category requires being able to count sets of objects

                • How Many in Total? hard

                  Answering 'how many?' requires the cardinality principle

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

                • One-to-one counting hard

                  Counting objects to answer 'how many?' requires one-to-one correspondence

            • Sorting Data into Categories soft

              Data representation formats (pictograms, tally charts) support organising data

              • How Many in Total? soft

                Counting data in categories requires understanding cardinality

                • One-to-one counting hard

                  Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

              • Pictograms and tally charts (age 6+) hard

                Organising and representing data requires data, tally, frequency, and category vocabulary

              • Sorting into categories hard

                Organising data in categories builds on classifying and counting objects in categories

                • Comparing groups: more or fewer soft

                  Sorting categories by count benefits from ability to compare quantities

                  • Counting objects to 20 soft

                    Counting a set helps when comparing groups, but younger children (GB age 4) can compare using matching without formal counting to 20

                    • How Many in Total? hard

                      Answering 'how many?' requires the cardinality principle

                      • One-to-one counting hard

                        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

                    • One-to-one counting hard

                      Counting objects to answer 'how many?' requires one-to-one correspondence

                • Counting objects to 20 hard

                  Counting objects in each category requires being able to count sets of objects

                  • How Many in Total? hard

                    Answering 'how many?' requires the cardinality principle

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

                  • One-to-one counting hard

                    Counting objects to answer 'how many?' requires one-to-one correspondence

          • Likelihood Language hard

            Conducting probability experiments and describing results requires knowing the language used to describe likelihood

        • Unit fractions hard

          Expressing probabilities as unit fractions (1/6, 1/2, 1/3) requires prior knowledge of unit fractions from the Fractions domain

          • Fractions of amounts hard

            Finding fractions of discrete sets extends finding fractions of shapes/quantities

            • Finding halves and quarters (age 5+) hard

              Working with 1/4, 2/4, 3/4 extends from Y1 understanding of quarters

              • What Is a Half? hard

                Understanding quarters extends from understanding halves — both are equal parts but quarters requires dividing into 4

                • Division as equal sharing hard

                  Finding a half requires equal sharing into 2 groups — a division concept

                  • Subtraction as taking away or separating hard

                    Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                    • How Many in Total? hard

                      Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                      • One-to-one counting hard

                        Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

            • What Is a Half? hard

              Working with fractions extends from Y1 understanding of halves

              • Division as equal sharing hard

                Finding a half requires equal sharing into 2 groups — a division concept

                • Subtraction as taking away or separating hard

                  Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                  • How Many in Total? hard

                    Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                    • One-to-one counting hard

                      Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

            • Division as equal sharing soft

              Finding fractions of quantities uses equal sharing (division)

              • Subtraction as taking away or separating hard

                Division as equal sharing/grouping requires understanding subtraction as taking away/separating

                • How Many in Total? hard

                  Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                  • One-to-one counting hard

                    Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

            • Fraction Notation hard

              Writing fractions like 1/3 and 3/4 requires knowing numerator and denominator

          • Division as equal sharing soft

            Finding 1/4 of 12 objects connects to division as sharing equally

            • Subtraction as taking away or separating hard

              Division as equal sharing/grouping requires understanding subtraction as taking away/separating

              • How Many in Total? hard

                Understanding subtraction as taking away requires knowing numbers represent quantities (cardinality)

                • One-to-one counting hard

                  Cardinality principle builds on one-to-one correspondence — you must count correctly to know the last number tells 'how many'

        • Equally Likely Outcomes hard

          Using fractions to represent probability only makes sense for equally-likely outcomes, so the equally-likely concept must come first

          • Ordering Likelihoods hard

            Understanding what 'equally likely' means is a specific case of comparing likelihoods that requires the general comparison skill first

            • Likelihood Language hard

              Comparing and ordering likelihoods requires first knowing the vocabulary of likelihood (certain, likely, unlikely, impossible)

      • Comparing fractions soft

        Expressing probability as fractions, decimals, and percentages requires comparing and ordering fractions — a skill built in the Fractions domain

    • Equally Likely Outcomes hard

      The probability formula only applies to situations with equally likely outcomes — this concept must be secure first

      • Ordering Likelihoods hard

        Understanding what 'equally likely' means is a specific case of comparing likelihoods that requires the general comparison skill first

        • Likelihood Language hard

          Comparing and ordering likelihoods requires first knowing the vocabulary of likelihood (certain, likely, unlikely, impossible)

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