Probability

• How to Calculate Probabilities in a Markov Chain
  Calculate Markov-chain probabilities using transition matrices, initial distributions, matrix powers, path products, and hitting-probability equations.
• How to Use the Normal Distribution to Find Probabilities
  Learn to use the normal distribution by standardizing values to z-scores, finding cumulative probabilities, and checking whether the model fits.
• How to Find Probabilities for a Uniform Distribution
  For a continuous uniform distribution on [a, b], find an event’s probability by dividing its overlap length with [a, b] by b − a.
• What Is the Expectation of a Function Symmetric About Zero?
  A function symmetric about zero is even, so its expectation is not generally zero; symmetry makes expectations of integrable odd functions vanish.
• How to Find the Expected Squared Distance in a Random Walk
  For a random walk on a graph, calculate expected squared distance by weighting each squared graph distance by the probability of reaching that vertex.
• How to Calculate Probabilities in Minesweeper
  Calculate a Minesweeper square’s mine probability by counting all mine layouts consistent with clues and the total mine count.
• Closed Form for a 2D Random Walk Probability Distribution
  For a square-lattice walk with four equally likely directions, count paths to (x, y) and divide by 4^n; the formula also identifies unreachable points.
• Conditional Mean and Variance of an Exponential Random Variable
  For an exponential random variable conditioned to exceed a threshold, memorylessness gives a shifted exponential: mean threshold plus 1/rate and variance 1/rate squared.
• How to Calculate the Probability of a Shared Birthday
  Calculate the chance that at least two people share a birthday by finding the probability that all birthdays differ, then subtracting it from 1.
• How to Compute the MGF of a Compound Poisson Distribution
  Compute a compound Poisson MGF by conditioning on the Poisson count: the result is exp(lambda times (the jump MGF minus one)).
• Continuity Correction vs. Confidence Interval: What’s the Difference?
  A continuity correction improves normal approximations to discrete probabilities; a confidence interval estimates an unknown parameter. Here is when each applies.
• How to Find Probabilities in a Box of Red and Black Balls
  For a draw without replacement, use combinations to find the probability of a given number of red and black balls, or adapt the method for replacement.
• When Is a Random Variable Almost Surely Constant?
  A real random variable independent of itself must be almost surely constant; see the proof, a Bernoulli example, and other useful criteria.
• What Are the Expected Value and Standard Deviation of Dice Rolls?
  For a fair six-sided die, the expected roll is 3.5 and the standard deviation is about 1.708; for n independent rolls, use the scaling rules.
• Expected Value of 1/(X + 1) for a Poisson Distribution
  For X ~ Poisson(λ), E[1/(X+1)] = (1−e^−λ)/λ. See the series derivation, a worked example, and the special case λ = 0.
• Expected Value of a Geometric Distribution by First-Step Analysis
  Use first-step analysis to derive the mean of a geometric waiting time, then see how the answer changes when you count failures instead of trials.
• What Is the Expected Value of a Poisson Distribution?
  For a Poisson random variable with parameter λ, the expected value is λ. See the derivation, a worked rate example, and common mistakes.
• How to Prove the Law of Total Expectation
  The law of total expectation says E[Y] = E[E[Y|X]]. Learn its proof from conditional expectation's definition, with discrete, continuous, and worked numerical examples.
• How to Find the Median and Modes of a Discrete Uniform Distribution
  For a discrete uniform distribution, every support value is a mode. The median is the middle value for odd support size and spans two middle values for even size.
• How to Find a Probability Density Function (PDF)
  Learn how to derive a probability density function from a CDF, identify a known distribution, or transform a random variable, then check its support and integral.
• Law of Iterated Expectations: A Worked Example
  The law of iterated expectations says to average a conditional mean over the variable you conditioned on; a weighted two-bag example shows each step.
• How to Write the Not-Independent Symbol in LaTeX
  Use the centernot package to place a slash across the double-perpendicular independence symbol, with a complete LaTeX example and spacing guidance.
• Expected Value of 1/(1 + X) for a Geometric Random Variable
  For a geometric random variable X, E[1/(1+X)] is -p ln(p)/(1-p) on support starting at zero; starting at one gives a different formula.
• What Is the Tower Property of Conditional Expectation?
  The tower property says that conditioning first on finer information and then on coarser information is the same as conditioning directly on the coarser information.
• What Does π Mean in Statistics?
  In statistics, π often denotes a population proportion, event probability, prior distribution, or mixture weight; its exact meaning comes from context.
• Given the Mean and Standard Deviation, How Do You Find a Probability?
  Convert values to z-scores and use the standard normal table to find P(X < b), P(X > a) and P(a < X < b) from a mean and standard deviation, with examples.
• What Is the Standard Deviation of a Dice Roll?
  The standard deviation of one fair six-sided die is √(35/12) ≈ 1.708. See the derivation, the formula for any die, and how it scales for sums and averages.