AdelaideX: MathTrackX: Probability

• How to understand and interpret probabilities depending on the context

• The difference between a discrete random variable and a continuous random variable

• How to calculate probabilities for a range of everyday scenarios

• How to calculate the expected value, variance and standard deviation of random variables

• The effects of linear changes of scale and origin on the mean and the standard deviation

• How to calculate quantiles of normal distribution.

Week 1: Basic probability

• Understand the product and sum principles of counting and apply them to solve a variety of counting problems
• Recognise when a problem involves a permutation or combination and apply the corresponding formula to count the number of possible results
• Understand the concept of probabilities associated with the outcome of a random experiment
• Understand the concept of an event as a set of outcomes within the sample space and be able to compute the probabilities associated with events
• Use and interpret Venn diagrams to describe compound events
• Understand the concepts of independent, mutually exclusive and conditional events and be able to recall and apply the appropriate formula to solve problems in probability
• Recall and apply the addition law of probability to solve a variety of problems involving compound events, and
• Use and interpret tree diagrams for problems involving conditional probabilities.

Week 2: Discrete random variables

• Understand the concept of a random variable and be able to identify/distinguish discrete and continuous random variables
• Understand the probability distribution associated with a discrete random variables and its properties
• Interpret a variety of representations of discrete random variables including histograms and probability tables
• Recall the definition of the mode, median, expected value, variance and standard deviation of a discrete random variable
• Calculate the mode, median, expected value, variance and standard deviation of a wide range of discrete random variables
• Understand the relation between the properties of a discrete random variable which is a function of another, particularly when that function is linear
• Recognize both Bernoulli Binomial experiments and the properties of the random variables associated with each • Apply knowledge of discrete random variables to solve a range of problems.

Week 3: Continuous random variables

• Understand the concept of a continuous random variable and its corresponding probability density function
• Verify when a function is a valid probability density function
• Recall the definition and formula for the mode, median, expected value, variance and standard deviation of a continuous random variable
• Calculate the mode, median, expected value, variance and standard deviation for continuous random variables which have a reasonably simple probability density function
• Understand the relation between the properties of a continuous random variable which is a function of another
• Recognise the properties of the normal distribution
• Recall and apply the 68:95:99.7 rule to calculate probabilities of a normal random variable over intervals which are bounded by appropriate intervals
• Apply the standard normal transformation to find probabilities and quantiles associated with normal random variables more generally
• Use technology to find probabilities and quantiles associated with normal random variables more generally.
• Use derivative rules and trigonometric functions, exponential functions and logarithmic functions to solve a variety of problems.

Week 4: Assessment

There is a timed exam.