In the previous lesson, we discussed the big picture of the course without really delving into why the study of probability is so vitally important to the advancement of science. Let's do that now by looking at two examples.

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Example 2-1
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Suppose that the Penn State Committee for the Fun of Students claims that the average number of concerts attended yearly by Penn State students is 2. Then, suppose that we take a random sample of 50 Penn State students and determine that the average number of concerts attended by the 50 students is:

\(\dfrac{1+4+3+\ldots+2}{50}=3.2\)

that is, 3.2 concerts per year. That then begs the question: if the actual population average is 2, **how likely** is it that we'd get a sample average as large as 3.2?

What do you think? Is it likely or not likely? If the answer to the question is ultimately "not likely", then we have two possible conclusions:

**Either:**The true population average is indeed 2. We just happened to select a strange and unusual sample.**Or:**Our original claim of 2 is wrong. Reject the claim, and conclude that the true population average is more than 2.

Of course, I don't raise this example simply to draw conclusions about the frequency with which Penn State students attend concerts. Instead I raise it to illustrate that in order to use a random sample to draw a conclusion about a population, we need to be able to answer the question "how likely...?", that is "what is the probability...?". Let's take a look at another example.

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Example 2-2
Section* *

Suppose that the Penn State Parking Office claims that two-thirds (67%) of Penn State students maintain a car in State College. Then, suppose we take a random sample of 100 Penn State students and determine that the proportion of students in the sample who maintain a car in State College is:

\(\dfrac{69}{100}=0.69\)

that is, 69%. Now we need to ask the question: if the actual population proportion is 0.67, **how likely** is it that we'd get a sample proportion of 0.69?

What do you think? Is it likely or not likely? If the answer to the question is ultimately "likely," then we have just one possible conclusion: The Parking Office's claim is reasonable. Do not reject their claim.

Again, I don't raise this example simply to draw conclusions about the driving behaviors of Penn State students. Instead I raise it to illustrate again that in order to use a random sample to draw a conclusion about a population, we need to be able to answer the question "how likely...?", that is "what is the probability...?".

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Summary
Section* *

So, in summary, why do we need to learn about probability? Any time we want to answer a research question that involves using a sample to draw a conclusion about some larger population, we need to answer the question "how likely is it...?" or "what is the probability...?". To answer such a question, we need to understand **probability**, **probability rules**, and **probability models**. And that's exactly what we'll be working on learning throughout this course.

Now that we've got the motivation for this probability course behind us, let's delve right in and start filling up our probability tool box!