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