Sets, Counting and Distributions

Probability starts by counting outcomes, and it stops working once there are too many to list. Two things replace listing: counting rules that give a total without writing it out, and distributions that cover a whole family of situations.

Why bother with set notation?

Because set notation says exactly which outcomes you mean, and ordinary English does not: "tea or coffee" excludes having both, but the union does not. A set is a collection of distinct objects, and the universal set is everything under discussion.

The intersection $A\cap B$A ∩ B is what lies in both sets at once. The complement of A is everything in the universal set that is not in A, and B is a subset of A when every element of B is also in A. See Set Notation, Venn Diagrams and Complements and Subsets.

To count a union, add the two totals and subtract the overlap once: $n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)$n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Anyone in both sets was counted twice.

A probability is then a count of regions divided by the count of the whole, and a condition narrows which regions count. See Counting a Union, Probability in Set Notation and Conditional Probability from Tables.

Some infinities are strictly larger: the decimals between 0 and 1 cannot be listed even in an endless list. See Cantor's Diagonal: Bigger Infinities.

How do you count without listing?

By multiplying choices instead of writing out the possibilities. Choices made one after another multiply; a choice of one option or another adds.

Three starters and four mains give 12 two-course meals; three starters or four mains, one course only, gives 7. See Adding and Multiplying Choices.

Arranging n different things in a row can be done in n! ways: n choices for the first position, n − 1 for the second, down to 1. There is one way to arrange nothing, so 0! = 1. See Factorials.

Permutations or combinations?

Ask whether reordering the same selection produces a different outcome. If it does, count permutations; if it does not, count combinations.

A permutation fills r positions from n candidates in order: nPr = n!/(n − r)!. A combination has no order: nCr = n!/(r!(n − r)!). Eight runners give 336 orderings of three medals but only 56 teams of three. See Permutations and Combinations.

When items repeat, divide by the factorial of each repeat: BANANA arranges in 6! ÷ (3!2!) = 60 ways. When two people must sit together, glue them into one item and multiply by the 2 ways they can sit inside. See Repeats and Restrictions.

A triangle is an unordered choice of three vertices, so 10 points with no three in a line give 120 triangles. See Counting with Geometry.

What is a random variable?

A random variable attaches a number to every outcome of an experiment: instead of "two heads" you write X = 2, and X becomes something you can average. A discrete variable takes values you can list; a continuous one takes any value in a range. See Random Variables.

A probability distribution lists every value X can take with the probability of each. See Probability Distributions.

The expected value E(X) is the mean of that distribution: multiply each value by its probability and add. It is a long-run average per trial, not a prediction of one trial. See Expected Value.

Variance measures spread as the expected squared distance from the mean, usually computed as $E\left(X^2\right)-{\left[E\left(X\right)\right]}^2$E(X²) − [E(X)]². Squaring stops distances above and below the mean canceling. See Variance.

Changing units changes the two differently: E(aX + b) = aE(X) + b, but Var$\left(aX+b\right)=a^2$(aX + b) = a²Var(X), because variance is measured in squared units. See Transforming a Random Variable.

What is the binomial distribution for?

For counting the successes in a fixed number of trials that each have the same two outcomes. One such trial is a Bernoulli trial: success with probability p, independent of the trial before. See Bernoulli Trials.

Write X ~ B(n, p) for the number of successes in n of those trials, and $P\left(X=r\right)=nCr\times p^r\times {\left(1-p\right)}^{n-r}$P(X = r) = nCr × p^r × (1 − p)^(n − r). See The Binomial Distribution.

The mean and variance need no summing: E(X) = np and Var(X) = np(1 − p), so twenty tosses of a fair coin have mean 10 and variance 5. See Binomial Mean and Variance.

Check four conditions first: a fixed number of trials, two outcomes each, the same p every time, and independent trials. Independence is the one that fails: twenty people from the same household are not twenty independent trials. See Modeling with a Binomial.

How does the normal distribution work?

It describes a continuous quantity clustered symmetrically around a mean, and probability is read as area under the curve, whose total is 1. See Continuous Variables.

The curve has two parameters: the mean sets its position, the standard deviation $\sigma$σ its width. Two standard deviations either side hold about 95% of the values, and three hold 99.7%. See The Normal Distribution.

There is a different curve for every mean and standard deviation, so no book could tabulate them all. Standardize instead: $z=\frac{\left(x-\mu \right)}{\sigma }$z = (x − μ)/σ turns every normal distribution into the same standard normal curve, with mean 0 and standard deviation 1. See Standardizing.

A z-table gives the area to the left of a z-score: subtract it from 1 for a right-hand tail, subtract the smaller from the larger for a band, and use symmetry for a negative z. An inverse normal question runs the table backward: the top 10% starts where 90% of the area lies to the left, so z = 1.28, then undo the standardizing with $x=\mu +z\sigma$x = μ + zσ. See Reading the Z-Table, Reading Normal Probabilities and Inverse Normal.

Sums and differences of independent normal variables are normal again, and their means add. See Combining Normals.

The mistakes worth naming

• Using a permutation where order does not matter. The same three people are the same committee however they were listed.
• Using a binomial when p changes. Drawing cards without replacement changes p at every draw.
• Subtracting variances for a difference. For independent variables, Var(X − Y) = Var(X) + Var(Y).
• Asking for P(X = 5) from a normal distribution. An exact value has probability 0, so ask about an interval.

Where this leads next

Everything here describes a distribution you assume. Deciding from data which assumption is defensible is the sampling and inference guide. The counting rules feed the tree diagrams in the probability guide, and the combination numbers are the coefficients in the binomial theorem.

Learn this properly in the app

Math Challenge teaches each of these as an illustrated lesson with a worked calculation and practice questions. Start with Sets and Counting or Distributions.