Find conditional probability from a frequency table

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find conditional probability from a frequency table.  Here's our problem statement: In an experiment, college students were given either four quarters or a $1 bill, and they could either keep the money or spend it on gum.  The results are summarized in the table.  Compare Parts A through C below.

Part A

OK, Part A asks us to find the probability of randomly selecting a student who spent the money given that the student was given four quarters.  Conditional probability is calculated by looking at the probability of both events occurring and then dividing that by the probability of just the one event occurring.  To do this, we can actually leave our data here in the table; we don't have to dump our data into any third party software.  I'm actually going to put the data into Excel because Excel makes it kind of nice for our purposes.  And I'll show you how nice it can be here in a moment.

You can work the problem in StatCrunch, but the problem with working in StatCrunch is that for this type of application, StatCrunch is actually pretty clunky and awkward.  You can do it, but because it's so awkward to use StatCrunch, I'm actually going to use Excel.  So we'll open up Excel here.

All right, here's my data here in Excel — whoa!  Oh, wow.  Yeah, OK, so that's a different problem.  Let's actually move you over here because I think — yes, that actually matches.  Hey, hey, where you going?  Stupid window!  Go where I want you to go!  Yeah, let's actually put you right here.  Yes, let's get rid of you.  What are you moving back here for?  Ah, ah, why can't you be where I want you to be?  Why are you trying to think for me?  Do what you're told!  OK, here we go.  Now we're ready.

So now I have the data here.  So we wanted to find the probability of randomly selecting a student who spent the money given that the student was given four quarters.  OK, well, first the probability of both — the probability that you both spent the money (which means you purchased the gum) and that you the student was given four quarters.  So four quarters is this row.  So the point where they both intersect is here at the 26.  So that's going to go on top.

So we're going to take 26, and we're gonna divide that by the probability that the student was given the four quarters.  Well, how many students were given the four quarters?  This is where Excel makes it really nice, because all I have to do is just select the cells for that given category (given the four quarters) and then down here Excel actually sums up everything really nice for me.   The sum is 42.

I can get the same thing in this calculator just by using the parentheses.  See, there's my 42 all the same.  But either way you work it, this is why I say you don't really need any third-party software; you just need to know which numbers to put in, and then you can actually get it out.  There's your probability.  I'm asked to round to three decimal places.  Fantastic!

Part B

Now, Part B asks us to find the probability of randomly selecting a student who spent the money given that the student was given a one dollar bill.  OK, so same calculation, but just slightly different numbers.  So again, we want to get the students who spent the money and was given the $1 bill.  So you spent the money and you were given the one dollar bill.  That's gonna be the 19.  So 19 divided by — and then how many were given the one dollar bill?  47.  There's my probability.  Again I'm asked to round to three decimal places.  Nice work!

Part C

And the last part, Part C, asks, “What do the preceding results suggest?”  Before we look at our answer options, let's go ahead and look at what the preceding results actually are.  So here we've got two probabilities.  They're both probabilities for students who spend the money, but the condition is the thing that's different between them.  So in this case, the probability that's higher is associated with having the four quarters as opposed to the dollar bill.  So what we see here is that, if you're gonna spend the money, it's more probable you'll spend it if you actually have four quarters as opposed to the dollar bill.

OK, let's go back and look at our answer options and see which one matches what we've just learned.  Answer option A says, “A student was more likely to have spent the money than to have kept the money.”  Well, we didn't do any probabilities with keeping the money, just the probabilities of students who spent the money.  So that's not going to be right.  Answer option B says, “A student given four quarters is more likely to have spent the money than a student given a $1 bill.”  And yes, that's what we learned just a moment ago.  So I'm going to select that.

But just to make sure, let's check the other answer options before we check our answer.  Answer option C says, “A student given a $1 bill is more likely to have spent the money.”  No, we don't — that’s not what this is saying right here; it's actually the reverse.  And then answer option D says, “A student was more likely to be given four quarters than a one dollar bill.”  Well, the problem statement doesn't say anything about how students were selected for each of the categories or groupings.  It just says that they were selected.  We presume that it's random, but we don't really know because there's nothing about it that's said in the problem statement.

So it looks like answer option B is the one we're going to go with.  Well done!

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