Level 7: Random variables and expected value

Probability · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

Until now, probability has been about single events: will it happen or not? In this level chance produces numbers: the total on two dice, the score on a guessed quiz, the number of rolls until a six. A random variable and its distribution describe all those numbers at once; the expected value says what they average in the long run, and the standard deviation says how far they stray. Then come the famous distributions (binomial, geometric, hypergeometric, uniform and a first look at Poisson), each with a proof of where its formula comes from and a guide to choosing the right one.

Lesson	You will be able to
Random variables and their distributions	Build the distribution table and histogram of a random variable, check it adds to 1, and read probabilities off it
Expected value	Find the long-run average of a random variable, decide whether a game is fair, and use the rules for adding expected values
Spread of a random variable	Find the variance and standard deviation, compare risky and safe games, and use the rules for sums of independent variables
The binomial distribution	Recognise a binomial situation, derive and use \(P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}\), and find its mean \(np\)
The geometric distribution	Find the chance of waiting \(k\) trials for a first success, prove the expected wait is \(\tfrac1p\), and explain "no memory"
The hypergeometric distribution	Find probabilities when sampling without replacement, and judge when the binomial is a good enough approximation
Uniform distributions and choosing a model	Use the uniform and Poisson distributions, choose the right model for a situation, and get answers from a calculator or spreadsheet

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