Transcript Probability

PROBABILITY ESSENTIALS
• Concept of probability is quite intuitive; however, the
rules of probability are not always intuitive or easy to
master.
• Mathematically, a probability is a number between 0 and
1 that measures the likelihood that some event will occur.
– An event with probability zero cannot occur.
– An event with probability 1 is certain to occur.
– An event with probability greater than 0 and less than
1 involves uncertainty, but the closer its probability is
to 1 the more likely it is to occur.
Probability
•Probability plays a key role in statistics
•Probability theory has two basic building
blocks.
–The more fundamental is the generating
mechanism, called the random experiment, that
gives rise to uncertain outcomes
For example, selecting one part for inspection
from an incoming shipment is a random
experiment having two elements of uncertainty:
which particular item gets picked and the quality
of that part.
Probability
• The second structural element of probability
theory consists of the random experiment’s
outcomes, referred to as events
For example: during a quality-control
inspection, the usual events of interest for
tested items are “goods” and “defective”
Probability
• Elementary Events
An example: If we are interested only in the particular
upside showing face from a coin toss, then we have
Coin toss sample space = {head, tail} (showing face)
A preliminary step in a probability evaluation is to catalog
the events that might arise from the random experiment.
Such a listing is made up of elementary events, which are
the most detailed events of interest.
The immediate concern when finding a probability is
properly identifying the event. Any subset of a sample
space is called event. By subset we mean any part of a
set.
Sample Space
• In statistics, a set of possible outcomes of an
experiment is called sample space. Sample
spaces are usually denoted by the letter S.
• A sample space might be portrayed as a
list, as above, or in some other convenient
form.
• Sometimes it is conceptually helpful to use
a picture
Probability
The probability of an event (happening or
outcome) is the proportion of times the
event would occur in a long run of
repeated experiments.
For example, if we say that the probability is
0.78 that a jet from New York to Boston
will arrive on time, we mean that such
flights arrive on time 78% of the time.
Example
• If records show that 294 of 300 ceramic
insulators tested were able to withstand a
certain thermal shock, what is the
probability that any one such insulator will
be able to withstand the thermal shock?
Solution: Among the insulators tested,
294/3000 = 0.98 were able to withstand
the thermal shock
Basic Definitions of Probability
• Long-run Frequency: If a perfectly balanced
coin is tossed many times without bias toward
either side, we should obtain a head in about half
the tosses.
• Thus, the long-run frequency of “head” is 0.50.
We may then assume that 0.50 is the probability
of “head” expressed symbolically as
Pr[head] = 0.50
Objective Probability
• The previous value is an objective probability,
and there should be no agreement about how to
find its value.
• Objective probabilities can often be found,
as above, through deductive reasoning
alone.
• We don’t actually have to toss the coin to
reach the answer since we have no
reason to believe that the head side will
show any more or less than half the time.
Probability for an event
If elements are equally likely, Probability for
an event can be deduced directly,
Pr[event] = size of the event/ size of sample
space
Certain Event
• An outcome bound to occur is a certain
event
• It has a probability of 1.
Impossible Event
• An outcome having zero probably is called
an impossible event, since it cannot occur.