Statistics

A geometric distribution gives the probability that the first success happens on trial k, when independent trials each have the same probability of success p. The mean number of trials until the first success is 1/p.

A binomial distribution gives the probability of getting exactly k successes in n independent trials, when each trial has the same probability of success p. Its mean is np, and its standard deviation is the square root of np(1 − p).

Combining random variables means finding the mean and standard deviation of a sum or difference like X + Y or X − Y. The mean of a sum or difference is the sum or difference of the means. For independent random variables, the variances add, for sums and for differences.

The expected value of a random variable X is its long-run average: the mean you would get if the random process were repeated many times. To find the expected value of a discrete random variable, multiply each value by its probability and add the products.

A discrete random variable is a variable whose value is a number set by chance, with values you can count, like 0, 1, 2 and so on. Its probability distribution lists every possible value with its probability. Each probability is between 0 and 1, and all of them add to 1.

Bayes’ theorem reverses a conditional probability. If you know how likely B is when A happens, it finds how likely A is when B happens: P(A | B) = P(A) · P(B | A) ÷ P(B). In practice, draw a tree diagram, multiply along each branch, and divide the path you want by the total for B.

Two events are independent if knowing that one happened doesn’t change the probability of the other. In symbols, P(A | B) = P(A). For independent events, the multiplication rule is P(A and B) = P(A) · P(B). To test for independence, check whether either of these equations is true.

Conditional probability is the probability of an event when you already know that another event happened. It is written P(A | B) and read “the probability of A given B.” The formula is P(A | B) = P(A and B) ÷ P(B). In a two-way table, you just use the row or column for B as the whole.

The addition rule in probability finds the chance that event A or event B happens: P(A or B) = P(A) + P(B) − P(A and B). You subtract the overlap so it isn’t counted twice. If A and B are mutually exclusive (they can’t both happen), the overlap is 0, so P(A or B) = P(A) + P(B).

The basic probability rules say that every probability is between 0 and 1, the probabilities of all outcomes in the sample space add to 1, and the probability that an event does not happen is 1 minus the probability that it does. The law of large numbers says long-run relative frequencies settle near the true probability.