Probability models
Choose binomial and normal models, interpret their parameters and translate the event before calculating its probability.
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Before you begin
Choose binomial and normal models, interpret their parameters and translate the event before calculating its probability.
Learning goals
- Choose and use a binomial probability model
- Calculate an event probability in a specified binomial model
- Calculate a conditional probability in a specified binomial model
- Calculate the expectation of a specified binomial count
- Calculate the variance of a specified binomial count
- Recover the unique binomial parameters from exact moments
- Find the least trial count satisfying a binomial probability threshold
- Represent random variables, expectations and variances
- Choose and use a normal probability model
- Calculate an event probability in a specified normal model
- Calculate a conditional probability in a specified normal model
- Recover a normal boundary from its specified tail probability
- Recover a normal mean from a probability and known variance
- Recover a normal variance from a probability and known mean
- Recover the unique normal mean and variance from two probabilities
- Calculate a linear combination's expectation from specified moments
- Calculate a linear combination's variance with justified dependence conditions
- Calculate a probability for a specified combination of normal variables
Syllabus statements covered
- Include: concept of discrete random variables; concept of binomial distribution B(n, p), its use as a probability model and conditions for suitability; mean and variance without proof. The concept of discrete random variables is for teaching and learning only.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
- Include: concept of discrete random variables; concept of binomial distribution B(n, p), its use as a probability model and conditions for suitability; mean and variance without proof. The concept of discrete random variables is for teaching and learning only.
- Include: concept of continuous random variables; concept of normal distribution as a continuous probability model with mean and variance; use of N(μ, σ²); standard normal distribution; finding P(X < x₁) or related probabilities given x₁, μ, σ, and finding a relationship between x₁, μ, σ given a probability; symmetry and properties of the normal curve; problems using E(aX + b) and Var(aX + b), and E(aX + bY) and Var(aX + bY) for independent X and Y. Exclude normal approximation to binomial distribution. The concept of continuous random variables is for teaching and learning only.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Binomial models and calculator tailsChoose a binomial model, translate exact and conditional tails, recover parameters and combine component and box counts.
- Normal models and linear combinationsInterpret normal areas and inverse tails, recover parameters, and compare transformed or independent measurements under a model.