Basic Concepts of Probability CEE 431/ESS465

Basic Concepts of Probability Sample spaces and events Venn diagram  A Sample space,  Event, A

Intersection of sets  A Sample space,  B A B U Basic Concepts of Probability Sample spaces and events

Union of sets  A Sample space,  B A B U Basic Concepts of Probability Sample spaces and events

The probability of an even is represented by a number greater than or equal to zero but less than or equal to 1: 0 < P[A] < 1 The probability of an event equal to the entire sample space is 1 P[  ] = 1 The probability of an event representing the union of two mutually exclusive events is equal to the sum of the probabilities of the two events P[A U B] = P[A] + P[B] Basic Concepts of Probability Axioms of probability

P[A U B] = P[A] + P[B] - P[A B]  A Sample space,  B A B U U Basic Concepts of Probability Probabilities of events

P[A | B] =  A Sample space,  B U P[A B] P[B] Basic Concepts of Probability Conditional probability So, P[A B] U = P[A | B] P[B]

P[A] = P[A] = P[A|B 1 ]P[B 1 ] + P[A|B 2 ]P[B 2 ] + … + P[A|B N ]P[B N ] B1B1 B2B2 B3B3 B4B4 B5B5 A U P[A B 1 ] + U P[A B 2 ] + … + U P[A B N ] Basic Concepts of Probability Conditional probability

Basic Concepts of Probability Random Variables Quantities that can take on many values Discrete random variables - finite number of values Number of borings encountering peat at a site Date of birth Continuous random variables - infinite number of values Undrained strength of a clay layer Weight

Basic Concepts of Probability Continuous Random Variables Distribution of values described by probability density function (pdf) that satisfies the following conditions: The probability that X is between a and b is equal to the area under the pdf between a and b The probability that X is between a and b is equal to the area under the pdf between a and b

Basic Concepts of Probability Continuous Random Variables Distribution of values can also be described by a cumulative distribution function (CDF), which is related to the pdf according to

Basic Concepts of Probability Statistical Characterization of Random Variables Distribution of values can also be characterized by statistical descriptors Mean Variance Standard deviation Standard deviation

Basic Concepts of Probability Common Probability Distributions Uniform distribution f X (x) = 0 for x < a 0 for x > b 1/(b - a) for a < x < b a b fX(x)fX(x)FX(x)FX(x) xabx 1.0

Basic Concepts of Probability Common Probability Distributions Normal distribution fX(x)fX(x)FX(x)FX(x) xx 1.0 x x

Basic Concepts of Probability Common Probability Distributions Standard normal distribution Mean = 0 Standard deviation = 1 Values of standard normal CDF commonly tabulated

Basic Concepts of Probability Common Probability Distributions Standard normal distribution Mapping from random variable to standard normal random variable Compute Z, then use tabulated values of CDF

Basic Concepts of Probability Common Probability Distributions Example: Given a normally distributed random variable, X, with x = 270 and  x = 40, compute the probability that X < 300 Looking up Z = 0.75 in CDF table, F Z (0.75) = 1 - F Z (-0.75) =

Basic Concepts of Probability Common Probability Distributions Lognormal distribution fX(x)fX(x)fX(x)fX(x) ln xx 1.0 ln x