Finding probability from a frequency table and identifying disjoint events

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 a probability from a frequency table and identify disjoint events.  Here's our problem statement: Use the data in the following table, which lists drive through order accuracy of popular fast food chains.  Assume that orders are randomly selected from those included in the table.  If one order is selected, find the probability of getting an order from Restaurant A or an order that is accurate.  Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint events?

Part 1

OK, the first part of this problem is asking us to find a probability.  And when you're looking at a frequency table, like what we have here, I find it's easier to work the problem in Excel.  So I'm going to go ahead and open my data in Excel.  Here we go.  Oh, that's something else.  OK, here we go with my data here in Excel.  Let's move this over a bit so we can see better what's actually going on.

OK, so here's my data in Excel, and let's list some numbers here.  We could actually punch these numbers that we're going to calculate in a calculator.  But since we're in a spreadsheet and they're made for calculations, we'll just put all of our numbers right here, and that'll just make it easy.  So we were asked for the probability from getting from Restaurant A or we could have the order to be accurate.  OK, and then of course we've got to include events that you might double count.  And then that's going to give us the part that we want to calculate.

Then we got to get the whole, because the probability is just going to be the part divided by the whole.  So now let's go and grab our numbers so we can do our calculations.  First the probability of getting something from Restaurant A.  So the part here --- that's going to be this column here --- so if I select every number in that column, notice down here at the bottom, Excel already calculates the sum for me.  So here I got 368.

If you want, you can also use the functions in Excel.  I'm going to calculate a sum function here.  Select those same cells and I get the same number.  But since it calculates the sum for us, it's easier to just select and then put the number in.  Orders that are accurate will be in this column here, so it's going to be 963.

Double count --- these are the ones that are included in both Restaurant A and the orders that are accurate.  So where those two, where the row and the column intersect, that's going to be the number that's actually double counted, which here is the 335 because we counted this when we were counting Restaurant A, but we also counted this when we're counting the orders that were accurate.  So that number got double counted.  We got to subtract that out.

Now we're ready to calculate the part for our probability calculation.  So I press the equal sign, come up here, select Restaurant A, I'm going to add it to the orders that are accurate, and then I want to subtract out, the amount that's double counted.  I press Enter, and there's my part.

The whole is just everything in the table.  So I select every number in the table and the outcome is 1098.  Then the probability is just going to be the part divided by the whole.  And there's my probability.  Notice the answer field here doesn't list a percent sign, which means the answer is wanted in decimal form.  And we're instructed to round to three decimal places as needed.  So I'm just going to take that number, round it to three decimal places.  Fantastic!

Part 2

And now the second part of the problem asks, "Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint events?"  Well, disjoint means mutually exclusive.  In other words, there is no overlap or intersection between those two events.  They're not occurring at the same time.  So the easiest way is to just look at your table.  Are there any areas of overlap?

And the answer is yes.  Remember that we had to take out the 335 that was double counted.  The reason why it was double counted was because it's in both of those events.  It's an accurate order and it's from Restaurant A.  So it's an area of overlap.  And because of that, the events are not disjoint.  So here we're going to say the events are not disjoint because it is possible to receive both an accurate order and get an order from Restaurant A.  Fantastic!

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