Download Chapter 12: Introducing Probability The idea of probability

Chapter 12: Introducing Probability
The idea of probability
Randomness and probability
Phenomenon Random: If individual outcomes are uncertain but there is nevertheless a regular
distribution of outcomes in a large number of repetitions.
Probability: The chance of an event occurring.
The proportion of times the outcome would occur in a very long series of repetitions.
(1) Relative Frequency Approximation of Probability
Conduct (or observe) a procedure, and count the number of items event A
actually occurs.
P (A) =
#of times A occurred
#of times procedure was repeated
Probability and Outcomes that are not Equally Likely
(2) Classical Approach to Probability
Assume that a given procedure has n different simple events and that
each of those simple events has an equal chance of occurring.
P (A) =
# of ways A can occur
# of different simple events
Probability and Outcomes that are Equally Likely
(3) Subjective Probability
P (A), the probability of eventA, is estimated by using knowledge
of the relevant circumstance.
Probability models
Event: the outcome or the results of a procedure. A subset of the sample space.
Sample Space (S): the set of all possible simple events.
Probability Model: The mathematical description of a random phenomenon consisting
of two parts: a sample space S and a way of assigning probabilities
to events.
Notation for Probability
Event: an outcome that is usually denoted by a capital letter; A, B, C.
A, B, C: Specific events
P: A probability
P(A): The probability of event A occurring
The complement of event A: all outcomes
in which event A does not occurs.
A: The complement of event A
P(A): The probability of the complement of event A
Probability rules
(1) The probability of an impossible event is 0.
(2) The probability of an event that is certain to occur is 1.
(3) 0 ≤ P (A) ≤ 1
(4) Rounding Off Probabilities
When expressing the value of a probability, either give the exact fraction or decimal
or round off final decimal results to three significant digits.
An event is unlikely
If its probability is very small, such as 0.05 of less.
Finite (Discrete) probability models
Finite probability model: A probability model with a finite sample space is called finite.
Proper reasonable Models
Binomial Distribution
Poisson Distribution
Continuous probability models
Continuous probability model: A continuous probability model assigns probabilities as areas
under a density curve. The area under the curve and above any range of values is the probability
of an outcome in that range.
Proper reasonable Models
Normal Density Curve (Normal Distribution)
Right-Skewed Density Curve
Left-Skewed Density Curve
Uniform Density Curve (Uniform Distributions)
Uniform Distributions:
1. Its values are spread evenly over the range of possibilities.
2. The graph of a uniform distribution in a rectangular shape.
3. The area under the graph of a probability distribution is equal to 1.
Random variables
1. A random variable is a variable whose value is a numerical outcome of a random phenomenon.
2. The probability distribution of a random variable X tells us what values X can take and how
to assign probabilities to those values.
3. Random variables that have a finite list of possible outcomes are called finite (discrete).
4. Random variables that can take on any value in an interval, with probabilities given as areas
under a density curve, are called continuous.