Finding probability from a frequency table for selections with and without replacement

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 probability from a frequency table for selections with and without replacement.  Here's our problem statement: Use the data in the following table, which lists drive through order accuracy at popular fast food chains.  Assume that orders are randomly selected from those included in the table.  If two orders are selected, find the probability that they are both from Restaurant D.  (a) Assume that the selections are made with replacement.  Are the events independent?  (b) Assume that the selections are made without replacement.  Are the events independent?

Part A

OK, so we're asked to find a probability, but we're asked to find it under two different conditions.  Part A is asking us to find the selections made with replacement.  So what we're going to do first is take this frequency table, and I find it easier to find probabilities with frequency tables by working in Excel.  So I'm going to dump this data into Excel and, OK, there's my --- I don't know what it's doing here.  Whoops.  OK.  Let's see here.  I want to resize this.  Alright, let's move you over.  Behave, behave.  What are you doing?  Come back over here.  OK.  Let's bring that down a bit.  All right, so now I can see more of what's going on.

Here's my data in Excel and we're asked to make two orders.  And for each of these orders we're going to need to calculate a part and a whole.  And we take the part divided by the whole, and that gives us a probability for that part.  So there's two selections.  We're going to select the first order and then we're going to select a second order.  So now we just need to grab our numbers to fill in the table and run our calculations.  You could follow along with a calculator if you want, but the beauty about Excel is that being a spreadsheet, it's made for performing calculations inside.  So it's just really easy to just put the calculations here in Excel, as you'll see in a moment.

We're making selections with replacement.  So what we take out, we're going to put back in.  So the part for the first order --- so we're looking at the probability that the selection is made from Restaurant D.  So what's the part for Restaurant D?  The part that contains Restaurant D is right here in the table --- the 149 plus the 16 in which, if you look down here at the bottom, Excel will automatically sum the cells that we select.  So here we've got 165.  The whole is just everything in the table.  So here that gives us 1114.  And then the probability is just going to be the part divided by the whole.  So there's the first probability.

Now the probably for the second order, the selection is being made with replacement.  So this number is not going to change.  The part is still going to be 165.  The whole doesn't change.  It's still 1114.  So we've got the same calculation.  Now I could type it in again, or I could just come back here to this cell that has the formula I want in it.  And notice when I put the cursor over this little rectangle here on the bottom right corner of the selected cell, notice how my cursor changed to a plus sign.  Once it does that, it says it's ready for copying the contents of that cell over to neighboring cells.  So now that I've got that little plus sign for my cursor, I hold down the left button on my mouse, and I can drag that over to the cell I want a copy to, then release the button on my mouse.  And lo and behold, look, it copied the same formula, just transferred it over.  So I don't have to retype it.  That could be really useful in a lot of situations.  So a little free trick there for you for working in Excel if you didn't know that.

Now I've got the probabilities for each of the orders.  We want the probability for everything together.  So we want the probability that both of the orders are from Restaurant D.  That means the probability of the first order is from Restaurant D and the probability of the second order is from Restaurant D.  And means we multiply.  So I'm going to take this first probability and multiply it by the second probability.  And there's my total overall probability.  Put that in bold text if I want to highlight the difference there a little bit.

I need to round to four decimal places, so I can either look at it here and round it or I can actually come here into Excel.  Whoops.  Wrong way.  I just get down until I get to four, and it will round it automatically for me.

Now are the events independent?  Well, the events are independent because there's no connection between the two samplings.  So, you know, what I get for the second sample is not influenced by what I get from the first sample because I'm making selections with replacement.  Everything comes back in.  So yes, the events are independent because choosing the first order does not affect the choice of the second order.  Nice work!

Part B

Now Part B asks us to run the same calculations, only this time selections are made without replacement.  So looking at the first order, there's going to be the same numbers that we had previously for the part and the whole.  So that doesn't change.  But now what changes is the second order.  The part for the second order is going to change because what we took out from the first selection is not put back in to be selected again.  That means we've got one less order to select from, so I have to take one away.  Once I press Enter, notice how everything updates for me.  That's the beauty of doing this in Excel.  I don't have to run through the same calculations again.  Once I set it up, the computer updates everything for me automatically.  It's a beautiful thing.

So here's my new probability, and you can see that it's not much different from what we had before.  And that's because our sample size is relatively large.  We've got 1114.  If we had a much smaller sample size, then we would see more of a difference between the probabilities of selecting with and without replacement.  But that difference gets minimized the larger your sample size becomes.

Here we are asked again if the events are independent.  Here the events are not independent because we had to make a change in that second order because it was influenced by the fact we took one out for the first order and did not put it back in.  We're selecting without replacement.  So here are the events are not independent because choosing the first order does affect the choice of the second order.  Well done!

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