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Chapter 4 Notes

# Chapter 4 Notes - 4.1 and 4.2 Which is your favorite...

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4.1 and 4.2 Which is your favorite chapter so far? Chapter 1 ? Chapter 2 ? Chapter 3 ? Note how each of these chapters is very different from one another. In fact, you may have been wondering why in the world did we go through Chapter 2 , and then go through the craziness of Chapter 3 only to find out that it seems like Chapter 3 has nothing at all to do with Chapter 2 . You may have heard the old saying, “opposites attract”. Well, that’s what we will see here today as each of you will now witness a very special wedding. In this wedding, we unite two chapters in holy matrimony, and those two chapters happen to be Chapter 2 and Chapter 3 . Chapter 2 . Do you take Chapter 3 and promise that you will stand by it until death do you part?” “I do.” Chapter 3. Do you take Chapter 2 and promise that you will stand by it until death do you part?” “I do.” “So, we unite Chapter 2 and Chapter 3 , and together, they become Chapter 4 . Let’s see why Chapter 4 is like a combination of Chapter 2 and Chapter 3. Let’s suppose you want to do a survey of how many children each family has in your neighborhood. Knock, knock, knock. “How many children do you have?” “I have 3 children”. So, let’s make a chart like this: x = number of children. x P(x) 0 1 2 3 1 4 5 6 At this time, we have one family in our group. We can say P(a family has 3 children) = 1, and all the other probabilities are 0. Knock, knock, knock. “I have 1 child”. So, at this time, we have x P(x)

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0 1 0.5 2 3 0.5 4 5 6 Now, we go to a 3 rd house. “I have 1 child”. So, now, we have x P(x) 0 1 2/3 2 3 1/3 4 5 6 Now, you might think, “There is a probability of .95 that I have seen this example before somewhere.” You in fact would be right. Remember “Relative Frequency Approximation of Probability”? That was from Chapter 2 . “P(x)” is from Chapter 3 . So, there you go. We are seeing wedding bliss taking place. The wedding bliss in the form of the table above is what’s called a probability distribution. We already know that P(x) is always a number between 0, inclusive, and 1, inclusive. Symbolically, we write 0 ( ) 1 P x . Examining all the tables above, note that it always turns out that the sum of the P(x) column is always 1. The first table had one P(x) which was 1, and obviously the sum of 1 is 1. The 2 nd table above has 0.5 and 0.5, and 0.5 + 0.5 = 1. Then, note that 2/3 + 1/3 = 1. What we have discussed are in fact the 2 requirements of a probability distribution. Those 2 requirements are 1. 0 ( ) 1 P x 2. ( ) 1 P x = . By the way, x is called a random variable because the more people we add to our group, the probabilities “randomly” change. We do p. 194 #5. Note that all the P(x) listed are between 0 and 1, so requirement #1 is satisfied. .512 + .301 + .132 + .055 = 1 , so requirement #2 is satisfied. Therefore, the distribution listed is a probability distribution. To get an idea of what the probability distribution is trying to tell us, note that P(x = 0) = .512. This means that
the probability of picking ONE inmate who committed 0 prior sentences is .512.

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Chapter 4 Notes - 4.1 and 4.2 Which is your favorite...

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