Finding probability using the multiplication rule

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 using the multiplication rule.  Here's our problem statement: Multiple choice questions each have five possible answers, one of which is correct.  Assume that you guess the answers to three such questions.

Part A

OK, Part A says, "Use the multiplication rule to find the probability of WCC, where C denotes a correct answer and W denotes a wrong answer."  To help us with our calculations, I'm going to pull up Excel, and I'm gonna run my calculations here in Excel.  So we're asked to find the probability of WCC, where the first question we answer is a wrong answer, and then we get a correct answer, and then we get a correct answer.  And then we're going to combine those probabilities together to get our final probability.

OK, now what's the probability of getting a wrong answer?  Well, there's five possible answers.  Only one's correct, so there's four that are wrong.  So the part over the whole --- four divided by five will give us 80%.  In fact, let's come over here and actually center things out.  So we got 80% for that probability.

Now if you can't do that in your head, you can always do it here in Excel.  Just type the equal sign, then put the part (4) divided by the whole (which is 5), press Enter, and voila!  I get the same answer that comes out.  So 80%.  To get a correct answer is 1 out of 5, which is going to be 20%.  And you can --- alternatively, if you want to calculate that out just like we did before, and you'll get the same number.

So the probability of WCC is the probability of getting the first one wrong and the probability of getting the second one correct and the probability of getting the third one correct.  So, and, when we're using probabilities, means multiplication.  So we're just going to multiply those three numbers together.

So there's two ways I can do that. I can press the equals sign, and then I can select each one of these cells in turn and multiply them together like you see here.  Press Enter, and there's my value.  The other way to do this is with the product function in Excel.  So to use that, I'm going to select --- take equal, then spell out product, open parentheses, I'm going to select these three cells that I want to put into the product, close the parentheses, Enter.  And notice I get the same number coming out.  So whichever way works for you, go for it.

Now notice here it says type an exact answer.  What that means is it wants a fraction.  What we have here as a decimal, so how do we get the fraction from the decimal?  How do we convert that out?  Well, if we notice here with the first question that we answered, there's five possible selections and the same for the second and the same for the third.  If I just copy this formula down, that means I've got 125 total possible outcomes in my sample set.  So out of 125, I've got 0.2% so I can actually multiply those together because the 125 is the whole.  When I multiply that by the part over the whole, I get just the part, just 4.  So the exact answer is going to be 4 divided by 125.  So I come over here --- 4 divided by 125.  Fantastic!

Part B

Now Part B says, "Beginning with WCC, make a complete list of the different possible arrangements of two correct answers and one wrong answer.  Then find the probability for each entry in the list."  We can see they have already listed that out for you because you've got the remaining answer spaces here to fill next to each possible outcome.  So they've made the list for you here.

But if we just --- instead of going back through all of this calculation again, which you can do if you want, or we can use a shortcut if we remember one of the properties of multiplication is that when you're multiplying different numbers together, the order doesn't matter.  You're going to come out with the same thing.  So if I take 1 times 2 times 3, I get six.  But if I take 1 times 3 times 2, I get six.  If I take 2 times 1 times 3, I get six.  3 times 1 times 2 is six.  It doesn't matter what the order is.  So we're going to get the same number come out for these combinations as we did for the earlier ones.  So I'm just going to put in the same number for each of these answer fields.  Good job!

Part C

Now Part C says, "Based on the proceeding results, what is the probability of getting exactly 2 correct answers when 3 guesses are made?"  And you may be tempted based on what we did with the last part to just put in the same answer that we gave here, and you would be incorrect to do that.  These are possible outcomes.  What this question is asking is for getting any one of these possible outcomes.  So it could be that we get this outcome here --- the first one wrong and the second two correct.  Or it could be that we get the first two correct and the last one wrong.  Or it could be that we get the second one wrong with the first and third correct.  We're going to get one of those outcomes: the first or the second or the third.

Or, when we're dealing with probability, means we add.  Or means addition.  So we're going to add 4 over 125 to itself three times.  And when we do that, we're going to get the answer that comes out here in our answer field.  So 4 plus 4 plus 4 is 12, so I want 12 over 125.  And notice I can't reduce that any further.  So there's my exact answer.  Nice work!

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