Computing Probabilities Using a Contingency Table
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Computing Probabilities Using a Contingency Table

Welcome to MVCC Learning Commons
Statistics Extras. Today I’ll be talking about computing probabilities using
contingency tables. At the bottom of the screen I have what’s called a
contingency table or what might also be referred to as a two-way table. It has
two different types of information and numbers that are cross-referenced
between them. This table is based off of a survey in which college students were
asked if they listen to music while studying. To begin, before we can start
answering questions we have to first find totals of the rows and the columns.
So I’m first going to find totals of the columns. So I’m going to make a note of
that at the bottom and I’m going to first find the total of the Listen
column. That number will be 268. The total of the Do Not Listen column is 132. Now I’m
going to go and find the totals of the rows. So for Male that will be 199 and
for Female that will be 201. Now in order to do anything with the problems we need
to find a complete table total. So if we add either the rows or the columns I
will get my table total, which in this case will be 400. So now if we want to begin by answering
some questions, we’re going to take these numbers to find specific probabilities.
So for our first question we’re looking for the probability that
the student is male. So in order to do this we’re going to look strictly for
the Male students, which is right across the top, right in here and we’re looking
for the total here so we’re looking at the end. So it’ll be 199 over everybody
in the chart so it’s going to be out of 400.
So it will be the same thing for those that
listen. We’re again looking for everybody that listens to music, which in this case
is the Listen column, and we’re again looking for the total. So this will be
268 out of 400. And just check with your instructor as to what form they want
these answers in. Whether they want them in unreduced fractions or as decimals
rounded to a specific location. You just want to be aware of that when answering
these questions. So if we look at another set we might have something that’s a
little more complicated. So question 3 says find the probability that the
student is female and does not listen to music. So if we’re looking at an “and”
situation we’re looking for someplace where these two intersect in the chart.
So we’re looking for Female and Do Not Listen which is right here. So
where they intersect is going to be 54. So this one will be 54 again out of
everybody in the chart which is 400. For the next one, this one will be male
students or they listen to music that one’s different.
This is an “or” here. So when we’re looking for “or”s and let me clear off
the marks in my chart. If we’re looking for “or”s we’re not looking for where they
intersect here we’re looking for everybody. So all of the Males are here and all that Listen to music are here. So
all the males are 199, all that listen to music is 268 but do you see where they
cross? We have 121 that are counted in both the Males and those that Listen to
music. We don’t want to count them twice. So in this case we’re going to subtract
the 121 so they’re not counted twice. So when we add and subtract these numbers
we get 400 sorry 346 and now this number is going to again be out of everybody in
the chart which is out of 400. So now if we look at two more additional questions
these are specific types of questions. So number 5 states the probability that
somebody listens to music given that they’re female that’s what that line
means. So this line here means “given”. So it’s a vertical line not a slash or
anything it’s a vertical line. So in this case when you have a “given” that means
you’re looking out of a specific row or column total. So we always look to the
second one, so when we’re looking for students that Listen given that they are
Female we can mark up just our Female row in this case.
Now we’re only looking for the students that Listen out of that row, which is
going to be 147. Now because we are looking for just those that listen out
of only the Female students, our Female total is going to be our denominator
this time. So our denominator will now be 201. That’s the difference between these
questions and the prior questions. Anytime we have a “given” we are doing a
specific row or column total. Same thing with this last question. We have in this
case the probability student is male given that they listen to music. So again
let me clear out my chart and then I will re-mark it. Okay so now the “given” is the students
that listen to music, so we are going to highlight just the students that Listen to
music and now we’re only looking for the Males in that column so the males are
going to be 121 out of only the students that Listen to music which is 268. So
these are the type of problems that are the “given” problems. So I hope this
answers your questions on how to find probabilities from a contingency table
also known as a two-way table. If you have any additional questions don’t be
afraid to come see us in the Learning Commons. Thank you!

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