Probability
Basic Probability Concepts Probability Distributions Sampling Distributions
Probability
Basic Probability Concepts
Basic Probability Concepts
Probability refers to the relative chance that
an event will occur. It represents a means
to measure and quantify uncertainty.
0 probability 1
Basic Probability Concepts
The Classical Interpretation of Probability:
P(event) = # of outcomes in the event
# of outcomes in sample space
Example:
P(selecting a red card from deck of cards) ?
Sample Space, S = all cards Event, E = red card
thenP(E) = # outcomes in E = 26 = 1
# outcomes in S 52 2
Probability
Random Variables and Probability
Distributions
Random Variable
A variable that varies in value by chance
Random Variables
Discrete variable - takes on a finite, countable # of values
Continuous variable - takes on an infinite # of values
Probability Distribution
A listing of all possible values of the random variable, together with their associated probabilities.
Notation:
Let X = defined random variable of interest x = possible values of X P(X=x) = probability that takes the value x
Example:
Experiment:
Toss a coin 2 times.
Of interest: # of heads that show
Example:
Let X = # of heads in 2 tosses of a coin (discrete)
The probability distribution of X, presented in tabular form, is:
x P(X=x) 0 .25 1 .50 2 .25
1.00
Methods for Establishing Probabilities
Empirical Method Subjective Method Theoretical Method
Example:
Toss 1 Toss 2
T T There are 4 possible
T H outcomes in the
H T sample space in this
H H experiment
Example:
Toss 1 Toss 2
T T P(X=0) = ?
T H Let E = 0 heads in 2 tosses
H T P(E) = # outcomes in E
H H # outcomes in S
= 1/4
Example:
Toss 1 Toss 2
T T P(X=1) = ?
T H Let E = 1 head in 2 tosses
H T P(E) = # outcomes in E
H H # outcomes in S
= 2/4
Example:
Toss 1 Toss 2
T T P(X=2) = ?
T H Let E = 2 heads in 2 tosses
H T P(E) = # outcomes in E
H H # outcomes in S
= 1/4
Example:Example:
The probability distribution in tabular form:
x P(X=x) 0 .25 1 .50 2 .25
1.00
Example:Example:
The probability distribution in graphical form:
P(X=x)1.00
.75
.50
.25
0 1 2 x
Probability distribution, numerical summary form:
Measure of Central Tendency:mean = expected value
Measures of Dispersion:variancestandard deviation
Numerical Summary Measures
Expected Value
Let = E(X) = mean = expected value
then
= E(X) = x P(X=x)
Example:
x P(X=x)
0 .25 1 .50 2 .25
1.00
= E(X) = 0(.25) + 1(.50) + 2(.25) = 1
Variance
Let ² = variance
then
² = (x - )² P(X=x)
Standard Deviation
Let = standard deviation
then = ²
Example:
x P(X=x)
0 .25 1 .50 2 .25
1.00
² = (0-1)²(.25) + (1-1)²(.50) + (2-1)²(.25)
= .5
= .5 = .707
Practical Application
Risk Assessment:
Investment A Investment B
E(X) E(X)
Choice of investment – the investment that yields the highest expected return and the lowest risk.
