Discrete random variables and their probability distributions, the requirement that probabilities sum to one, the use of statistical distributions to model real situations, and an introduction to the binomial and normal models.

What this dot point is asking

AQA wants you to understand discrete random variables, write and use a probability distribution, check that the probabilities sum to one, and select a suitable statistical distribution (such as the binomial or normal model) for a given situation, stating the assumptions you rely on. This topic provides the language and the total-probability rule that the binomial and normal distributions then build on.

Random variables

Capital letters such as $X$ denote the variable; lower-case $x$ denotes a particular value it can take. Distributions can be given as a table, a formula, or a description, and a frequent exam task is to convert between these forms.

The total probability condition

Distributions given by a formula

A distribution may be defined by a rule such as $P(X = x) = cx$ over a stated range of $x$. Substitute each allowed value, add the results, set the sum equal to one, and solve for the constant. Then you can read off or combine individual probabilities as needed.

Choosing a model

You select a distribution by matching its assumptions to the situation, and you should always state those assumptions because every model is an approximation.

• A fixed number of independent trials, each with two outcomes and the same probability of success, points to the binomial distribution $B(n, p)$.
• A continuous quantity that clusters symmetrically about a mean (such as heights, masses or measurement errors) points to the normal distribution $N(\mu, \sigma^2)$.
• A discrete count with no fixed upper limit, occurring at a constant average rate, would in later study point to other models, but at A-level the binomial is the main discrete model.

Reading and combining probabilities

Once a distribution is known, exam questions ask for probabilities of single values or of ranges. For a range, add the probabilities of the included values, watching the inequality symbols. The strict symbols $<$ and $>$ exclude the endpoint, while $\le$ and $\ge$ include it. The complement rule, $P(X \ge k) = 1 - P(X \le k - 1)$ for integer-valued variables, often saves work, and "at least one" is almost always fastest as one minus "none".

It is worth distinguishing a probability distribution from a frequency table of observed data. A probability distribution gives theoretical long-run proportions that must sum to one, whereas a frequency table records what actually happened in a sample and its entries sum to the sample size. Many marks are lost by treating observed frequencies as if they were probabilities without dividing by the total.