@ProbFact

Daily tweets on probability from @JohnDCook.

johndcook.com/blog/applied-probability/
Joined November 2009
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Putting probability to work johndcook.com/blog/applied-p…
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Bertrand's ballot theorem en.wikipedia.org/wiki/Bertra…

Bertrand's ballot theorem - Wikipedia

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A random variable is a measurable function on a probability space.
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Computing normal probabilities with a simple calculator johndcook.com/blog/2021/03/1…

Computing normal probabilities with a simple calculator

A simple approximation to the normal CDF for a calculator with basic functions and an exp key.

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“If the numbers are not random, they are at least higgledy-piggledy.” — RNG researcher George Marsaglia
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A probability space is a measure space such that the measure of the whole space is 1.
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'Probabilistic' comes from Latin. 'Stochastic' comes from Greek.
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Explaining probability to a jury johndcook.com/blog/explainin…

Explaining probability to a jury | Expert witness

A panel of jurors is unlikely to contain many people who understand probability, but understanding probability may be critical to a fair trial.

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The problem w/ trying to achieve reliability w/ multiple software implementations: they're not independent.
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A Short Introduction to Optimal Transport and Wasserstein Distance alexhwilliams.info/itsneuron…
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Uniformity increases entropy johndcook.com/blog/2025/06/2…

Uniformity increases entropy

Entropy increases as system states become more uniform and decreases as they become further from uniform.

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A random number generator may have excellent statistical properties and yet be unsuited for cryptography.
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Let τ be Kendall's tau and let ρ be Spearman's rho. Then -1 ≤ 3τ - 2ρ ≤ 1.
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'All probabilities are conditional on something, and to be useful they must condition on the right thing.' -- Frank Harrell
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Cosine power approximation to normal johndcook.com/blog/2020/07/2…

Cosine power approximation to normal

Cosine raised to a power makes a good approximation to a Gaussian probability density. The approximation improves as the power increases.

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The moment generating function of a random variable X is the two-sided Laplace transform of its density function f_X, with the sign of the argument flipped.
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The Mellin transform of the product of two independent random variables is the product of their Mellin transforms.
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Perfect and imperfect shuffles johndcook.com/blog/2026/01/0…

Perfect and imperfect shuffles

Shuffling a deck of cards works because ordinarily the process is imperfect. But with perfect shuffles, a deck will return to its original order.

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'There are only five probabilities the average human can handle: 99 percent, one percent, 100 percent, zero, and 50-50. That’s it.” — Richard Thaler
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Coefficient of variation = standard deviation / mean
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If A and B are independent causal factors leading to L, then L is called a collider. After conditioning on L, A and B are /not/ independent.
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