Probability, Random Variables, and Probability Distributions

A probability lies between 0 and 1. A sample space lists possible outcomes; an event is a collection of outcomes. Complements, intersections, and unions describe different combinations of events.

A probability lies between zero and one and describes a long-run pattern or a specified model of uncertainty. A complement contains outcomes not in the event. The addition rule subtracts overlap because outcomes in both events would otherwise be counted twice.

Conditional probability changes the reference group to outcomes satisfying the condition. In a two-way table, use the appropriate row or column total, not automatically the grand total. A high probability of a positive test among people with a condition does not imply a high probability of the condition among people with a positive test.

Independent events satisfy P(A|B) = P(A) when the condition is defined, or P(A ∩ B) = P(A)P(B). Knowing that one occurred does not change the probability of the other. Disjoint events cannot occur together. Two disjoint events with positive probabilities are not independent because one occurring rules out the other.

For a sequence of dependent events, multiply appropriate conditional probabilities. Sampling without replacement changes the composition of the remaining population. A sufficiently small sample relative to a large population can sometimes justify an approximate independence condition for later inference.

A simulation assigns outcomes to random numbers in proportions matching the model, repeats the process defining one trial, records the requested statistic, and repeats many trials. State how a random outcome is generated and what is counted as a success.

More repetitions reduce Monte Carlo variability but do not correct a simulation that models the wrong process. If sampling is without replacement, a simulation with independent repeated draws can be inappropriate unless the approximation is justified. A simulation estimate is not guaranteed to equal the theoretical probability exactly.

A discrete random variable has a countable set of possible values with probabilities summing to one. Its expected value is a probability-weighted average, not necessarily a possible individual outcome. Standard deviation describes variability around that mean in the variable’s units.

For a continuous model, probabilities correspond to intervals; a single exact point has probability zero even though that value is possible. Distinguish the distribution of individual observations from the sampling distribution of a statistic calculated from repeated samples.

A binomial model counts successes in a fixed number of trials with two outcome categories, independent trials, and a constant success probability. Define success before calculating. A count measured until some stopping condition is reached is not the fixed-trial binomial setup.

For X ~ Binomial(n,p), the mean is np and standard deviation is √[np(1 − p)]. Exact probabilities use the number of ways to choose successful trials times the probability of a particular success/failure sequence. Use cumulative probability carefully: “at least k” includes k, while “more than k” does not.

The sampling distribution of a sample proportion is centered at p under random sampling, with standard deviation √[p(1 − p)n] under independence. The sample mean is centered at μ, with standard deviation σ√n. These distributions concern repeated sample results, not individual people or measurements.

A central-limit approximation can make the sampling distribution of a mean approximately normal as sample size grows under suitable conditions. Severe skew, extreme tails, and dependence require care. Multiplying sample size by four halves the standard error; doubling sample size does not halve it.

“Among students who attended” defines a restricted denominator: only attendees belong in the reference group. P(A given B) divides the probability of both A and B by the probability of B. Reversing the condition changes the denominator and usually changes the answer. A positive screening result among people with a condition is not the same probability as having the condition among people with positive results.

A two-way table makes the denominator visible. For a conditional probability, divide the relevant intersection count by the total of the specified conditioning group. For a joint probability, divide by the overall total. Test independence by comparing a conditional probability with its unconditional counterpart, allowing for rounding when values are estimated.

For a discrete random variable, list numerical values and their probabilities. All probabilities must be nonnegative and sum to 1. The expected value is a weighted average, interpreted as a long-run mean over many repetitions; it does not have to be a possible individual outcome. Standard deviation measures the typical distance of outcomes from that mean, in the random variable’s units.

A binomial model requires a fixed number of trials, two outcomes per trial, independent trials, and the same success probability each time. When sampling without replacement, independence is approximate if the sample is at most 10% of the population. Identify what counts as success before using a calculator; “at least” includes the boundary value.

A sampling distribution describes a statistic across repeated samples of the same size, not the individual observations in one sample. The population distribution, the distribution of sample data, and the sampling distribution of a statistic are three distinct objects. A sample mean can have an approximately Normal sampling distribution even when the original population is not Normal, provided sample size and other conditions are appropriate.

Larger samples reduce standard error. For independent observations, the standard deviation of a sample mean is population standard deviation divided by the square root of sample size. Quadrupling the sample size halves that quantity; doubling the sample size does not. Random sampling centers the sample mean at the population mean, but no individual sample is guaranteed to land there.

Define events before choosing a rule. “A or B” includes outcomes in either event, including their overlap; subtract the overlap when adding probabilities. “A given B” restricts the reference group to outcomes where B occurs. The denominator changes, so P(A|B) need not equal P(B|A).

Independence means learning that one event occurred does not change the probability of the other. Disjoint events cannot occur together. Two events with positive probabilities that are disjoint are not independent: knowing one occurred makes the other impossible. Use the mathematical relationship rather than deciding from ordinary-language impressions.

A tree or two-way table can make conditional probabilities clearer. Label each branch with the probability appropriate to its preceding condition. Multiply along a path for a joint probability, then add mutually exclusive paths for a combined event. A simulation must reproduce the relevant probabilities and independence assumptions to model the situation.

A probability distribution lists possible values and associated probabilities, which must be nonnegative and total one. Expected value is a probability-weighted long-run average, not necessarily a possible single outcome. Standard deviation measures spread around that mean in the original variable’s units.

A binomial count requires a fixed number of trials, two outcome categories per trial, a common success probability, and suitable independence. Sampling without replacement changes probabilities; the 10% condition is a common approximation check when sampling from a finite population. Define “success” and the count X before using a calculator command.

For exactly k successes, use the binomial probability at k; for at most k, sum from zero through k. “More than k” excludes k. Sketch the included integers before entering a cumulative calculation. The binomial mean np and standard deviation √(np(1−p)) summarize the count, not the sample proportion.

A population distribution describes individual values; a sample distribution describes observed individual values; a sampling distribution describes a statistic across repeated samples of a fixed size. Increasing sample size narrows the sampling distribution of a mean or proportion under appropriate conditions. It does not necessarily make the population itself less variable.

For a sample proportion, the center is p and standard deviation is √(p(1−p)n) under independence. A normal approximation needs enough expected successes and failures. For a sample mean, the center is μ and standard deviation is σ√n. The central limit theorem concerns the sampling distribution of the mean, not a claim that raw data become normal.

To cut standard error in half, multiply sample size by four. This square-root relationship explains why precision improves more slowly than sample size. A biased collection method can remain centered away from the target even as this distribution becomes narrower.