Cheat sheet

When to use it: When the opposite is simpler, above all for "at least one": work out "none" and subtract it from 1.

When to use it: For "A or B". Take the overlap away once; for mutually exclusive events it is 0.

When to use it: When one event is known to have happened, or for the chance that both happen. Multiply the plain probabilities only for independent events.

When to use it: To turn the chance of a positive result given the condition into the chance of the condition given a positive result. Never leave out the base rate.

When to use it: The mean is the long-run average and the standard deviation the typical distance from it. Scaling and shifting a variable scales and shifts its mean; a shift leaves the spread alone.

The shop's units sold, k, and their probabilities, p, are in A2:A7 and B2:B7.

k = np.array([0, 1, 2, 3, 4, 5])
p = np.array([0.05, 0.15, 0.3, 0.25, 0.15, 0.1])

When to use it: Whole numbers in a known range, each equally likely: a fair die, a bookseller's 90 to 100 books a day. Not when real values cluster in the middle.

When to use it: A measurement equally likely anywhere in a range, when the range is all that is known. Every simulation starts from U(0, 1).

When to use it: The number of successes in a fixed number of independent trials, each with the same probability p. Not when sampling without replacement from a small group, or when there is no fixed n.

When to use it: Counts of events in an interval, arriving independently at a steady rate, with no fixed number of trials; rare events among many chances. Not when events cluster or the rate changes.

When to use it: Measurements that cluster symmetrically around a mean. Not for skewed data, values close to a bound, or questions about the tails of returns.

When to use it: To compare values measured on different scales, and to find a normal percentile as μ + zσ. Use the distribution's own μ and σ, not a handful of values'.

When to use it: To see why large counts look like a bell. Add the continuity correction, the half; with software at hand, compute the binomial exactly.

When to use it: The wait between events arriving independently at a steady rate λ, and random failures without wear. scipy takes the mean, scale = 1/λ; Excel takes the rate.

When to use it: A mean standardised with the sample's s in place of σ, from a roughly normal population; the critical values of t intervals. Close to the normal once the degrees of freedom are large.

When to use it: The sum of k squared standard normals: the distribution behind the sample variance and Student's t, and the tests of a variance in the Hypothesis Testing Lab.

When to use it: The mean of n independent values: its spread is σ/√n for every n, and its shape is close to normal once n is large. n ≥ 30 is a rule of thumb; skewed populations and questions about tails need more.

When to use it: To estimate μ and σ from a sample. Divide by n − 1 for s; numpy's std() divides by n unless ddof=1 is given.

The 10 wildfire damages are in B2:B11.

x = np.array([15, 80, 31, 104, 131, 46, 15, 62, 16, 36])

When to use it: When σ is known, as for a long-running machine, and the population is roughly normal or n is large. Excel's CONFIDENCE.NORM gives the margin of error, not the interval.

When to use it: When σ is estimated by the sample's s, from a roughly normal population when n is small. The 95 % describes the procedure: 95 % of intervals built this way contain μ.

The 10 wildfire damages are in B2:B11.

x = np.array([15, 80, 31, 104, 131, 46, 15, 62, 16, 36])