Extension Statistical analysis – Discrete probability distribution

This lesson comprises two (2) master classes focusing on:

• Conditional probability
• Multi-stage probability
• Discrete random variables
• Language of theoretical probability

Content:

MA-S1.1

• Understand and use the concepts and language associated with theoretical probability, relative frequency and the probability scale
• Solve problems involving simulations or trials of experiments in a variety of contexts
  • identify factors that could complicate the simulation of real-world events
  • use relative frequencies obtained from data as point estimates of probabilities
• Use arrays and tree diagrams to determine the outcomes and probabilities for multi-stage experiments
• Use Venn diagrams, set language and notation, including \( \bar{A} \) (or \( A^\prime \) or \( A^c \)) for the complement of an event /( A \), \( A \cap B \) for ‘\( A\ and\ B \)’, the intersection of events \( A\ and\ B \), and  \( A \cup B \) for ‘\( A\ or\ B \)’, the union of events \( A\ and\ B \), and recognise mutually exclusive events
  • use everyday occurrences to illustrate set descriptions and representations of events and set operations
• Establish and use the rules: \( P(\bar{A})=1−P(A) \) and \( P(A \cup B)=P(A)+P(B)−P(A \cap B) \)
• Understand the notion of conditional probability and recognise and use language that indicates conditionality
• Use the notation \( P(A|B) \) and the formula \( P(A|B)=\frac{P(A \cap B)}{P(B)},\ P(B) \ne 0 \) for conditional probability
• Understand the notion of independence of an event \( A \) from an event \( B \), as defined by \( P(A|B)=P(A) \)
• Use the multiplication law \( P(A \cap B)=P(A)P(B) \) for independent events \( A \) and \( B \) and recognise the symmetry of independence in simple probability situations

MS-S1.2

• Define and categorise random variables
  • know that a random variable describes some aspect in a population from which samples can be drawn
  • know the difference between a discrete random variable and a continuous random variable
• Use discrete random variables and associated probabilities to solve practical problems
  • use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable
  • recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes
  • examine simple examples of non-uniform discrete random variables, and recognise that for any random variable, \( X \), the sum of the probabilities is 1
  • recognise the mean or expected value, \( E(X)=\mu \), of a discrete random variable \( X \) as a measure of centre, and evaluate it in simple cases
  • recognise the variance, \( Var(X) \), and standard deviation (\( \sigma \)) of a discrete random variable as measures of spread, and evaluate them in simple cases
  • use \( Var(X)=E((X−\mu)^2)=E(X^2)−\mu^2 \) for a random variable and \( Var(x)=\sigma^2 \) for a dataset
• understand that a sample mean, \( \bar{x} \), is an estimate of the associated population mean \( \mu \), and that the sample standard deviation, \( s \), is an estimate of the associated population standard deviation, \( \sigma \), and that these estimates get better as the sample size increases and when we have independent observations