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Mathematics

Probability

By acquiring probabilistic reasoning skills, students avoid the confusion between probability and proportion and demystify certain false preconceptions about odds or chance, such as the bias associated with equiprobability, availability and representativeness. This prepares them to exercise their critical judgment in various situations.

In elementary school, students conducted experiments related to chance. They made qualitative predictions about outcomes using concepts related to outcome (certainty, possibility and impossibility) and the probability that an event will occur (more likely, just as likely and less likely). They listed the outcomes of a random experiment using tables or tree diagrams and compared the actual outcomes with theoretical probabilities.

In Secondary Cycle One, students go from using subjective, often arbitrary, reasoning to reasoning based on various calculations. They further develop the concept of probability of an event—the cornerstone in calculating probabilities—and are introduced to the language of sets. They learn to enumerate possibilities using different registers (types) of representation, to calculate probabilities and to compare experimental and theoretical probabilities. With this knowledge and skills, students are able to make predictions and informed decisions in various types of situations.

In Secondary Cycle Two, students continue to build on what they learned in Cycle One. They use the results of combinatorial analysis (permutations, arrangements and combinations) and add the ability to calculate probabilities in certain measurement contexts to their repertoire of knowledge and skills. Depending on the option, students learn to distinguish between subjective probabilities and experimental or theoretical probabilities. They interpret or distinguish between various relationships (e.g. the probability of an event and the odds for or the odds against). They use the concept of mathematical expectation to determine whether a game is fair or the possibility of a gain or a loss. Lastly, they analyze situations and make decisions based on conditional probability.

The following tables present the learning content associated with probability. By basing themselves on the concepts and processes targeted, students develop the three competencies of the program, which in turn enable students to better integrate the mathematical concepts and processes presented.

Understanding data from random experiments  
Student constructs knowledge with teacher guidance.
Student applies knowledge by the end of the school year.
 
Student reinvests knowledge.
Secondary
Cycle
One		 Cycle
Two
  1. Processing data from random experiments
 6 1 2 3 4 5
  1. Simulates random experiments with or without the use of technological tools
            
  1. Experiments with activities involving chance, using various objects (e.g. spinners, rectangular prisms, glasses, marbles, thumb tacks, 6-, 8- or 12-sided dice)
            
  1. In activities involving chance
    1. recognizes variability in possible outcomes (uncertainty)
            
    1. recognizes equiprobability (e.g. quantity of objects, symmetry of an object such as a cube)
            
    1. becomes aware of the independence of events (e.g. rolling dice, tossing a coin, drawing lots)
            
  1. Uses tables or diagrams to collect and display the outcomes of an experiment
            
  1. Compares the outcomes of a random experiment with known theoretical probabilities
            
  1. Distinguishes between prediction and outcome
            
  1. Conducts or simulates random experiments involving one or more steps (with or without replacement, with or without order)
          
  1. Identifies the type of random variable: discrete or continuous
           CST
    TS
    S
  1. Enumerates the possible outcomes of a random experiment using
 
    1. tables, tree diagram
           CST
    TS
    S
    1. networks, tables, diagrams, Venn diagrams
      Note : In developing their probabilistic thinking skills, students are introduced to the language of sets, which is considered to be a comprehension and communication tool.
         CST
    TS
    S
    1. geometric figures
           CST
    TS
    S
  1. Defines the sample space of a random experiment
         CST
    TS
    S
  1. Recognizes certain, probable, impossible, simple, complementary, compatible, incompatible, dependent, independents events
         CST
    TS
    S
  1. Distinguishes between mutually exclusive and non-mutually exclusive, and between dependent and independent events
           CST
  TS
    S
  1. Uses fractions, decimals or percentages to quantify a probability
          CST
    TS
    S
  1. Recognizes that a probability is always between 0 and 1
          CST
    TS
    S
  1. Predicts qualitatively an outcome or several events using a probability line, among other things
  
    1. certain, possible or impossible outcome
            
    1. more likely, just as likely, less likely event
            
  1. Uses factorial notation, if necessary
    Note : This notation may be introduced in CST.
             CST
  TS
    S
  1. Recognizes, depending on the context, different types of probabilities: experimental, theoretical, subjective
           CST
  TS
    S
  1. Defines or interprets the concept of odds/chance (odds for and odds against) (e.g. makes connections between odds and probabilities)
           CST
  TS
    S
  1. Defines or interprets the concept of mathematical expectation (e.g. makes connections between mathematical expectation and weighted mean)
           CST
  TS
    S
  1. Analyzing probability situations
6 1 2 3 4 5  
  1. Represents an event using different registers (types of representation))
           CST
    TS
    S
  1. Compares qualitatively the theoretical or experimental probability of an event occurring
            
  1. Distinguishes between theoretical and experimental probability
          
  1. Calculates the probability of an event
         CST
    TS
    S
  1. Calculates the probability of outcomes of random experiments related to situations involving arrangements, permutations or combinations
    Note : Calculations are based on reasoning, not on counting formulas. The terminology (permutation, arrangement, combination) may be introduced in the first year of Secondary Cycle Two.
           CST
    TS
    S
  1. Associates the type of probability to a situation: experimental, theoretical, subjective
           CST
  TS
    S
  1. Calculates probabilities, including geometric probabilities, in measurement contexts
          CST
    TS
    S
  1. Calculates conditional probabilities
           CST
  TS
    S
  1. Interprets probabilities and makes appropriate decisions
         CST
    TS
    S
  1. Chooses and applies the concept of odds/chance (odds for and odds against) or probability, depending on the context
          CST
  TS
    S
  1. Determines the odds for or odds against
           CST
  TS
    S
  1. Interprets and makes decisions with respect to the odds obtained
           CST
  TS
    S
  1. Calculates mathematical expectation
          CST
  TS
    S
  1. Modifies, if necessary, the parameters of a situation in order to make it fair, attain an objective or optimize a gain or loss
           CST
  TS
    S
  1. Interprets the resulting mathematical expectation and makes appropriate decisions
           CST
  TS
    S

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