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>> Marchetti: Today we're going to talk about conditional probability. We will start
with an example. We have our fictional ABC Company here.
>>And out of the 100 employees total, we have 90 that are right handed, and 10 that are left-handed.
>> To get started, we introduce some definitions. First a statistical experiment:
This is the random selection of a person or object to observe a characteristic.
For example, we could randomly select an employee from ABC Company, and
observe if they are right handed or left-handed.
>> Next, we define the sample space. Often denoted with a capital "S".
This is the set of all possible types of outcomes of the experiment.
For example, when we are selecting an employee from ABC Company, our sample
space consists of right handed and left-handed.
>> We often use a visual representation of the sample space. A box labeled "S"
represents everything that can happen. All the possible outcomes of the
experiment.
>> Next we define an event. This is a set of one or more of the possible outcomes of
the experiment. For example, our event could be that the selected
employee is left-handed.
>> We can show and event on the Venn diagram by taking a portion of the sample
space, usually a circle, and labeling that as the "event". In this case, as left-handed.
>> Next we introduce probability. This is the chance that the statistical experiment
results in a particular event, which we might call "E". The probability of E or P(E)
is equal to the number of outcomes that result
in E, divided by the total number of outcomes.
>> For example, the probability that randomly selecting an employee at ABC
Company results in a left handed employee is equal to the number of left-handed
employees divided by the total number of employees; or 10/100. Which equals .1
>> Conditional Probability is when we focus on part of the sample space, instead of
the entire sample space.
>> On the Venn diagram, when we can use all the options, everything in the box is
available to us. Alternatively, we could focus on the options that occur in a portion
of the sample space; inside some event. Such as the shaded area "B" in this Venn
diagram.
>> For example, if our ABC Company is split into three departments: Department A,
Department B, and Department C, then I could look for the probability of selecting
left handed employee if I only focused on the employees in Department B. We write
this in a special way. Instead of probability of just felt handed,
we now use a vertical line indicating a restriction or focus, and that focus is Department
B. Looking at the numbers, that within Department
B, there are five left handed employees, and a total of 20 employees.
The probability of left handed, when we focus on Department B, is 5/20, or .25
>> We can see this by going back to our visual representation. Originally, we had no
restrictions or focus; we had the entire sample space available to us. The left-
handed event takes up a certain portion of that sample space. So we can think of the
probability of left handed as the area that the left-handed event takes up, relative to
the area of the entire box. Or, we can translate that into the number of outcomes
that result in left handed relative to the total number of outcomes.
>> If we change our perspective and restrict ourselves, focusing on staying within
Department B, that means we have to stay within the shaded area of "B" inside the
sample space.
>> If we look for the probability of left handed when we are focused on department
B, we have to look to see how much area left handed takes up inside of "B" relative
to the area of "B".
>> The idea of where left-handed is inside of "B" is often referred to as the
intersection, which we denote with an upside-down "U".
Again, we can think of this in terms of area or relate it back to the number of
outcomes. The probability of left handed, when I am focused on Department B, is
the number of outcomes that are in that intersection where employees are left
handed AND in Department B, relative to the number of employees in Department B.
>> Finally, we look at the
classic definition of Conditional Probability, which doesn't
use area, and doesn't use number of outcomes, but uses probability.
As we have seen, those things can all be equivalent. The probability of left handed,
when we are restricted to or focused on Department B, is equal to the probability of
the intersection of left handed and Department B divided by the probability of
Department B.