The Foundation: Random Variables
At the heart of probability theory is the random variableβa process that assigns a number to each outcome of an experiment. The first crucial step in any problem is to identify whether you are dealing with a variable that is Discrete (counted) or Continuous (measured).
Discrete Random Variables (DRV)
A DRV can only take on a countable number of distinct values. Think in terms of counting whole items.
- βValues: Countable, distinct values (e.g., 0, 1, 2, …).
- βNature: Outcomes are counted.
- βRepresentation: Probability Mass Function (PMF), \(P(X=x)\).
- βExamples: Number of heads in 5 coin flips, shoe size.
Continuous Random Variables (CRV)
A CRV can take any value within a given range. Think in terms of precise measurements.
- βValues: Uncountable, infinite values within an interval.
- βNature: Outcomes are measured.
- βRepresentation: Probability Density Function (PDF), area represents \(P(a \le X \le b)\).
- βExamples: Height of a person, time to run a race.
Discrete Distributions
Explore distributions where outcomes are countable. We’ll look at general probability mass functions (PMFs) and the vital Binomial distribution.
Interactive PMF Explorer
A valid Probability Mass Function (PMF) must have probabilities that sum to 1. This explorer demonstrates finding an unknown constant \(k\) for the rule \(P(X=x) = k(x+c)\) and then calculates key statistics.
Binomial Distribution Simulator
The Binomial distribution models the number of successes in a fixed number (\(n\)) of independent trials. Use the sliders to see how \(n\) and \(p\) affect the distribution’s shape and centre.
Mean (\(np\))
Variance (\(np(1-p)\))
Calculate Probability
Continuous Distributions
Shift from counting to measuring. For continuous variables, probability is the area under a curve. Explore the ubiquitous Normal Distribution.
Normal Distribution Explorer
The Normal Distribution, or “bell curve,” is defined by its mean (\(\mu\)) and standard deviation (\(\sigma\)). See how these parameters affect its shape and use the calculator to find probabilities and z-scores.
Calculate Probability / Z-score
Statistical Inference
Inference is about using data from a small sample to make an educated guess about an entire population. The most common tool for this in VCE Methods is the confidence interval for a proportion.
Confidence Interval Calculator
A confidence interval provides a range of plausible values for an unknown population proportion, \(p\). Adjust the sample data and confidence level to see how they affect the interval and the margin of error.
Sample Proportion (\(\hat{p} = \frac{X}{n}\))
Margin of Error (\(M\))
Confidence Interval (\(\hat{p} \pm M\))
Interpretation
Based on this sample, we are 95% confident that the true population proportion, \(p\), is contained within the interval (0.000, 0.000).
Examination Mastery
Apply your knowledge. Here are key insights from VCAA examination reports to help you avoid common mistakes and secure maximum marks.
