>>Probability is a numerical measure of the likelihood that a specific event will occur.
There are two primary properties of probability. The first is that the probability of an event
lies on zero and one. That’s inclusive of zero and one though probabilities of zero
and one tend to be rare in the real world. The second property is that the sum of all
probabilities of all mutually exclusive events in the sample space of an experiment is one.
All possible events that could occur in the sample space are said to be collectively exhaustive.
There are three views of probability. The first is classical probability. Where we have
a more absolute notion of the universe of events and we can calculate the number of
ways to get our desired outcome. Here we know all the events in the sample space and we
know how to count the number of ways that our event of interest can occur. If we can
describe event A with classical probability then the probability of event A is the number
of ways that for A to occur divided by the total number of outcomes that could possibly
occur. For example let’s think about the probability of winning the lottery when the
lottery ticket requires us to pick six numbers from zero to nine. That is, let’s define
the event of interest as choosing the correct lottery number. We’ll discuss in detail
a little later on but as it turns out there are ten to the sixth or one million different
ways to pick a lottery ticket. So the total number of outcomes that could occur is one
million. If the lottery picks six numbers zero to nine at random, there’s only one
way to pick that winning lottery ticket. That is there’s only one-way for event A to occur.
The probability of A is in one out of one million. The classical view of probability
is what we need to find the probability of winning the lottery because we know exactly
how many ways that winning the lottery can occur and we know how many possibilities there
are in picking a lottery ticket. For another example of classical probability let’s consider
a state license plate number that is randomly assigned with six digits, letter letter letter,
and then number number number. Assume that all possible three letter combinations or
three letter combinations could appear on the license plate. Though there are probably
some inappropriate words or number combos that probably wouldn’t make it on a plate
but let’s not worry about that. Say we define event A as spelling ‘HAT’ on our license
plate. How many total license plates are possible? With twenty-six letters in the alphabet and
ten digits the number of ways to get letter letter letter and then number number number
is twenty-six times twenty-six times twenty-six times ten times ten times ten or seventeen
million five hundred seventy-six thousand. How many ways are there for the event, spelling
hat to occur? Naturally the three letters have to spell HAT then there are three numbers
remaining. One possibility is the plate HAT 175, another is HAT 549. In total there are
ten times ten times ten or one thousand possibilities to spell HAT on a license plate because there
are one thousand possibilities to arrange the three numbers that follow. The probability
of event A is one thousand divided by seventeen million five hundred seventy-six thousand
or one out of seventeen thousand five seventy-six. Again we know the number of ways for our particular
event A to occur and we know the total possible events that can occur. Event A is described
with a classical view of probability. This second view of probability is relative frequency.
With the relative frequency view of probability we describe events by using past history to
extrapolate about the occurrence of future events. We don’t know exactly how to calculate
the numerator or denominator of the probability calculation. We only have past experience.
Experiments are performed to generate data then the frequency of a desired outcome is
used to calculate the probability of that desired event happening again. When event
A is best subscribed with the relative frequency view the probability of event A is calculated
as the number of ways A has occurred in the past divided by the total number of outcomes
that we’ve witnessed in the past. Let’s say we define an event A as choosing a defective
part at random from a batch. We can’t explicitly calculate the numerator and denominator to
find the probability of a defective part. So we can’t use a classical probability
here. We look back at historic data. Find how many parts have been examined and how
many of those parts examined were defective. If eight hundred and seventy-four parts were
tested and thirteen of them were defective the probability of event A, choosing a defective
part at random is thirteen out of eight seventy-four or roughly, point oh one five. The third view
of probability is subjective probability. Where a probability is assigned to an event
based on some combination of subject judgment, experience, information, and belief. Here
the calculation of probability frequently involves experiments that cannot be repeated
and don’t have a finite and known set of outcomes. One example of subjective probability
involves sports gambling. Where some prior history is used but also some information
about players, venues, and weather conditions among others is accounted for when the likelihood
of a team winning is calculated. Another example is when a risk analyst tries to model the
likelihood of a terrorist event. Terrorist event in the US are rare and not much historical
data describing the occurrence of these events exists. So intelligence gathering near misses
and gut feel may go into calculating the probability that a terrorist event occurs. In this unit
we’ll talk about situations described with classical probability. Then in subsequent
units when we discuss random variables and their distributions, other views of probability
will come up. One last definition to cover here is marginal probability. A marginal probability
is simply the probability of a single event without any consideration of another event.
The notation for a marginal probability of event A is P(A) or the probability of A. All
the probabilities discussed in this video were marginal probabilities. If we perform
an experiment the outcome of two flips of a coin and if we define event B as observing
one or more tails then the probability of B is a marginal probability, a single event
without any consideration of other events.