A SHORT INTRODUCTION TO PROBABILITY

Because of the stochastic nature of genetics and evolution, we have to rely on the theory of probability.

TerminologyThe 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

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P( i) = 1i=1

n∑

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.

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P(A ∪ B) = P(A) + P(B)

Simple probabilitiesIf A and B are independent events, then the probability that both A and B occur is the intersection

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P(A ∩ B) = P(A)× P(B)

Simple probabilitiesExample: 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

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P(A |B) =P(A∩ B)

P(A)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.

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Np =n!

(n− r )!

Permutations

In how many ways can 8 CD’s be arranged on a shelf?

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Np =n!

(n− r )!

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n = 8

r = 8

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Np =8!

(8 − 8)!=

8!

1= 40,320

Permutations

In how many ways can 4 CD’s (out of a collection of 8 CD’s) be arranged on a shelf?

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Np =n!

(n− r )!

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n = 8

r = 4

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Np =8!

(8 − 4)!=

8!

4!=1,680

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.

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Nc =n

r

⎛

⎝ ⎜

⎞

⎠ ⎟=

n!

r!(n − r)!

Combinations

How many groups of 4 CDs are there in a collection of 8 CDs)?

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n = 8

r = 4

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Nc =n

r

⎛

⎝ ⎜

⎞

⎠ ⎟=

n!

r!(n − r)!

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Nc =8

4

⎛

⎝ ⎜

⎞

⎠ ⎟=

8!

4!(8 − 4)!=

8!

4!4!= 70

Probability DistributionThe 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

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P(i) =1

n

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

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P(k) =n

k

⎛

⎝ ⎜

⎞

⎠ ⎟pk 1− p( )

n− k

The binomial distribution

The mean and variance of a binomially distributed variable are given by

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μ =np

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V = npq

The Poisson distribution

Siméon Denis Poisson1781-1840

Siméon Denis Poisson1781-1840Poisson d’April

The Poisson distributionWhen the probability of “success” is very small, e.g., the probability of a mutation, then pk and (1 – p)n – k become too small to calculate exactly by the binomial distribution. In such cases, the Poisson distribution becomes useful. Let be the expected number of successes in a process consisting of n trials, i.e., = np. The probability of observing k successes is

The mean and variance of a Poisson distributed variable are given by μ = and V = , respectively.

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P(k) =λke−λ

k!

Normal Distribution

Gamma Distribution