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Learn about fundamental concepts of probability theory, its terminology, simple probabilities, conditional probabilities, permutations, and combinations, along with probability distributions like binomial and Poisson. Discover key principles and their real-world applications.
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A SHORT INTRODUCTION TO PROBABILITY Because of the stochastic nature of genetics and evolution, we have to rely on the theory of probability.
Terminology The possible outcomes of a stochastic process are called events. (A deterministic process has only one possible outcome.) A stochastic process may have a finite or an infinite number of outcomes. The probability of a particular event is the fraction of outcomes in which the event occurs. The probability of event A is denoted by P(A).
Terminology Probability values are between 0 (the event never occurs) and 1 (the event always occurs). Events may or may not be mutually exclusive. Events that are not mutually exclusive are called independent events.
The birth of a son or a daughter are mutually exclusive events. The birth of a daughter and the birth of carrier of the sickle-cell anemia allele are not mutually exclusive (they are independent events).
Terminology The sum of probabilities of all mutually exclusive events in a process is 1. For example, if there are n possible mutually exclusive outcomes, then
Simple probabilities If A and B are mutually exclusive events, then the probability of either A or B to occur is the union Example: The probability of a hat being red is ¼, the probability of the hat being green is ¼, and the probability of the hat being black is ½. Then, the probability of a hat being red OR black is ¾.
Simple probabilities If A and B are independent events, then the probability that both A and B occur is the intersection
Simple probabilities Example: The probability that a US president is bearded is ~14%, the probability that a US president died in office is ~19%, thus the probability that a president both had a beard and died in office is ~3%. If the two events are independent, 1.3 bearded presidents are expected to fulfill the two conditions. In reality, 2 bearded presidents died in office. (A close enough result.) Harrison, Taylor, Lincoln*, Garfield*, McKinley*, Harding, Roosevelt, Kennedy* (*assassinated)
Conditional probabilities What is the probability of event A to occur given than event B did occur. The conditional probability of A given B is Example: The probability that a US president dies in office if he is bearded 0.03/0.14 = 22%. Thus, out of 6 bearded presidents, 22% (or 1.3) are expected to die. In reality, 2 died. (Again, a close enough result.)
Permutations The number of possible permutations is the number of different orders in which particular events occur. The number of possible permutations are where r is the number of events in the series, n is the number of possible events, and n! denotes the factorial of n = the product of all the positive integers from 1 to n.
Permutations In how many ways can 8 CD’s be arranged on a shelf?
Permutations In how many ways can 4 CD’s (out of a collection of 8 CD’s) be arranged on a shelf?
Combinations When the order in which the events occurred is of no interest, we are dealing with combinations. The number of possible combinations is where r is the number of events in the series, n is the number of possible events, and n! denotes the factorial of n = the product of all the positive integers from 1 to n.
Combinations How many groups of 4 CDs are there in a collection of 8 CDs)?
Probability Distribution The probability distribution refers to the frequency with which all possible outcomes occur. There are numerous types of probability distribution.
The uniform distribution A variable is said to be uniformly distributed if the probability of all possible outcomes are equal to one another. Thus, the probability P(i), where i is one of n possible outcomes, is
The binomial distribution A process that has only two possible outcomes is called a binomial process. In statistics, the two outcomes are frequently denoted as success and failure. The probabilities of a success or a failure are denoted by p and q, respectively. Note that p + q = 1. The binomial distribution gives the probability of exactly k successes in n trials
The binomial distribution The mean and variance of a binomially distributed variable are given by
The Poisson distribution Siméon Denis Poisson 1781-1840 Poisson d’April
The Poisson distribution When the probability of “success” is very small, e.g., the probability of a mutation, then pkand (1 –p)n – k become too small to calculate exactly by the binomial distribution. In such cases, the Poisson distribution becomes useful. Let l be the expected number of successes in a process consisting of n trials, i.e., l = np. The probability of observing k successes is The mean and variance of a Poisson distributed variable are given by m = l and V = l, respectively.
