MathTrackX: Probability

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This course is part five of the MathTrackX XSeries Program which has been designed to provide you with a solid foundation in mathematical fundamentals and how they can be applied in the real world.

This course introduces probability and how it manifests in the world around us. Beginning with discrete random variables, together with their uses in modelling random processes involving chance and variation, you will start to uncover the framework for statistical inference.

Guided by experts from the School of Mathematics and the Maths Learning Centre at the University of Adelaide, this course will introduce discrete and continuous random variables and their applications in a variety of contexts.

Join us as we provide opportunities to develop your skills and confidence in applying mathematics to solve real world problems.

What you will learn

• 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.

Skills you learn

• Basic Math
• Stochastic Process
• Random Variables

Syllabus

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.