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The law of total probability states that if {B1, B2, ..., Bn} are mutually exclusive and exhaustive events, then P(A) = P(A and B1) + P(A and B2) + ... + P(A and Bn). Example: Given three equally likely production lines with defect rates of 1%, 2% and 3%, the total probability of getting a defective product is 1/3 * 1% + 1/3 * 2% + 1/3 * 3%.

Bayes' Theorem relates the conditional and marginal probabilities of events A and B: P(A|B) = (P(B|A) * P(A)) / P(B). Example: If the probability of disease A is 0.01 and the probability that a positive test is correct is 0.9, with the overall test positivity rate being 0.08, then the probability of having disease A given a positive test is (0.9 * 0.01) / 0.08.

The variance of a random variable measures the spread of its possible values. It is calculated by summing the squared difference between each possible value and the expected value, weighted by their probability. Example: For a fair die, variance is the sum of [((1-3.5)^2 + (2-3.5)^2 + ... + (6-3.5)^2) / 6].

A set of events is collectively exhaustive if at least one of the events must occur when an experiment is conducted. The sum of probabilities of all exhaustive events is 1. Example: The set of all possible outcomes on a dice (1 through 6) are collectively exhaustive.

Events A and B are independent if the occurrence or non-occurrence of A does not affect the probability of B occurring, so P(A|B) = P(A) and P(B|A) = P(B). Example: The result of one coin flip does not affect the result of another coin flip.

The number of ways to choose r items from n items without regard to order is $\binom{n}{r} = \frac{n!}{r!(n-r)!}$. Example: The number of ways to choose 2 fruits from a set of 5 is $\binom{5}{2} = 10$ ways.

The expected value of a random variable is the average of all possible values it can take, weighted by their probabilities. It is calculated by summing the product of possible values and their probabilities. Example: If a die has a 1/6 chance to land on any number from 1 to 6, the expected value is (1/6)*1 + (1/6)*2 + ... + (1/6)*6 = 3.5.

The standard deviation of a random variable is the square root of its variance and represents the dispersion of its possible values around the mean. Example: For a fair die, standard deviation is the square root of its variance value.

The addition rule states that the probability of either event A or event B occurring is P(A) + P(B) - P(A and B). Example: If there's a 30% chance it will rain (event A) and a 20% chance you'll find a dollar on the ground (event B), and the chance of both happening is 6%, then the chance of either event occurring is 30% + 20% - 6% = 44%.

The probability of the complement of event A (not happening) is 1 - P(A). Example: If the probability of it raining tomorrow is 30%, then the probability it will not rain is 1 - 0.3 = 70%.

The conditional probability of event A given event B is P(A|B) = P(A and B) / P(B). Example: If there's a 50% chance it will rain (event B) and a 20% chance you will be late to work given it rains (event A given B), then P(A|B) = 20%.

The number of ways to arrange n items in order is n! (n factorial). Example: The number of ways to arrange 3 books on a shelf is 3! = 6 different ways.

The multiplication rule states that the probability of both event A and event B occurring is P(A) * P(B), provided events A and B are independent. Example: If the chance of rolling a dice to get a six is 1/6, and flipping a coin to get heads is 1/2, then the chance of both happening is 1/6 * 1/2 = 1/12.

Disjoint or mutually exclusive events are events that cannot occur at the same time. For disjoint events A and B, P(A and B) = 0. Example: The events 'rolling a 3' and 'rolling an even number' on a dice are disjoint.