basic probability. theoretical versus empirical theoretical probabilities are those that can be...
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Basic Probability
Theoretical versus Empirical
Theoretical probabilities are those that can be determined purely on formal or logical grounds, independent of prior experience.
Empirical probabilities are estimates of the relative frequency of an event based by our past observational experience.
Theoretical Probability
Probability of A tossed coin landing on heads Drawing a spade from a poker
deck Observing a three when rolling a
die
Empirical Probability Empirical probability can be
subdivided into two categories: Objective versus Subjective
Probability that conception will result in twins (Objective)
Probability of an insurance applicant filing a claim (Objective)
Objective Probability
The previous examples can be considered objective in the sense that they are based on observations of past occurrences of events, under what are hopefully the same conditions that currently prevail.
Subjective Probability
Empirical in the sense that they are ultimately based on past observation
Subjective in the sense that the particular observation(s) upon which the particular probability estimate(s) are based, is not well defined, that is, a independent observer could not be instructed on how to arrive at the same probability
Subjective Probability What is the probability that a space satellite
will fall out of orbit and land on Tucson? What is the probability that a direct-
response advertisement will draw a profitable response?
What is the probability of extra-terrestrial life?
What is the probability that upon graduation, you will be offered a position on your first job interview?
Basic Probability Concepts
Probability Experiments Whenever we manipulate or make an
observation on our environment with an uncertain outcome, we have conducted an experiment.
Examples Taking an exam Tossing a coin Delivering a sales pitch
Probability Experiment Can be repeated many times
(at least in theory) Each such repetition is called a
trial When an experiment is performed
it can result in one or more outcomes, which are called events.
Sample Space The set of all possible outcomes of an
experiment is called the sample space, S, for the experiment
The outcomes in the sample space are called sample points
The outcomes forming the sample space must be mutually exclusive and exhaustive
The sample space and sample points depend on what the experimenter chooses to observe
Example – Toss a Coin Twice
Can record the sequence of heads (H) and tails (T), then S= {HH, HT, TH, TT}
Can record the total number of tails observed, then S= {0, 1, 2}
Can record whether the two tosses match (M) or do not match (D), so S= {M, D}
Exercise (Sample Spaces) Determine the sample space of the
following experiments: Toss a die and recording the number on the
top face Let a light bulb burn until it burns out Observe General Electric common stock and
recording whether it increased, decreased or remained unchanged during one market day
Record the sex of successive children in a three-child family
Events
An event, E, is a subset of the sample space and it denoted by
An event E is said to occur if the outcome of an experiment is an element of E
Consider the experiment of tossing a die once and recording the number on the top face.
The sample space, S= {1, 2, 3, 4, 5, 6}
SE
Example (Events) Some events associated with this
experiment are: E1={1} We observe a 1 E2={2} We observe a 2 E3={1,3,5} We observe an odd
number E4={1,2,3} We observe a number
less than 4.
Simple vs Compound Events
A simple event cannot be decomposed.
A compound event is an event that can be decomposed into other events.
E1 and E2 are simple events.
E3 and E4 are compound events.
Exercise Consider the experiment of flipping
a balanced coin three times. Determine the sample space for
the experiment List two events that correspond to
this experiment
Teminology Experiment Sample space Sample points Probability model Events Certain event Impossible event Mutually exclusive (disjoint) events
Discrete Sample Space A discrete sample space consists
of a finite number of sample points or a countable number of sample points.
Throughout Project 1, we will be concerned with finite discrete sample spaces.
Probability of an Event Given an event, we would like to assign it
a number, P(E), called the probability of E This number indicates the likelihood that
the event will occur. We can find this number by determining
the value of the ratio:
number of ways event can occur
total number of outcomes
Relative Frequency Suppose that we repeat the die tossing
experiment n times and notice that the event E1 occurs f times. We call the ratio f / n the relative frequency of the event after n repetitions.
If we repeat this experiment indefinitely and if the ratio f / n approaches a number, p, as n becomes larger and larger, then p is called the probability of the event.
Law of Large Numbers The more repetitions we take, the better
the approximationp f / n
This is sometimes referred to as the Law of Large Numbers, which states that if an experiment is repeated a large number of times, the relative frequency of the outcome will tend to be close to the probability of the outcome.
Summary of 20,000 Coin Tosses
Num of Tosses
Num. of Heads
Relative Freq.
n f f / n
10 8 .8000
100 62 .6200
1,000 473 .4730
5,000 2,550 .5100
10,000 5,098 .5098
15,000 7,649 .5099
20,000 10,038 .5019
Fundamental Properties
Upon analyzing the relative frequency concept, we see the following must hold:
1. Negative relative frequencies do not make sense
2. The relative frequency of the sample space must be 1
3. If two events are mutually exclusive, the relative frequency of their union must be the sum of their relative frequencies.
Fundamental Properties Cont.
1. For an event E, 0P(E) 12. P(S)=1, where S is the sample space3. P(E F)= P(E)+ P(F), where E and F
are mutually exclusive events4. P(E1 E2 … Ek)
= P(E1)+ P(E2)+…+ P(Ek), where the Ek’s are mutually exclusive.
Calculating P(E)
1. Define the experiment and clearly determine how to describe one simple event
2. List the simple events associated with the experiment and test each to be certain that they cannot be decomposed. This defines the sample space S.
Calculating P(E) Continued.
3. Assign probabilities to the sample points in S making certain that the Fundamental Properties for a discrete sample space are preserved.
4. Define the event, E, as a specific collection of sample points.
5. Find P(E) by summing the probabilities of the sample points in E.
Example A balanced coin is tossed three
times. Calculate the probability that exactly two of the three tosses results in heads.
Example
A balanced coin is tossed three times. Let E1 be the event that you observe
at least two heads. What is P(E1)? Let E2 be the event that you observe
at exactly two heads. What is P(E2)? Let E3 be the event that you observe
at most most heads. What is P(E3)? What can you say about E1 and E3
Basic Theorems of Probability
Let S be a discrete sample space. Theorem 1: P()=0, where is the
empty set. Theorem 2: For any two events E
and F in S, P(E F)= P(E) + P(F) - P(E F)
Theorem 3: If E is an event in S, then P(EC)= 1 - P(E)
Mutually Exclusive Two events are mutually exclusive if AB=.
If A and B are mutually exclusive, then
A B
( ) ( )P A B P A P B
Mutually Exclusive If no two events E1, E2, . . . , En can
happen at the same time, then
1 2 1 2..... ( ) ( ) ..... ( )n nP E E E P E P E P E