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371chapter2a - Chapter 2 Trials 2.1 Independent Identically...

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Chapter 2 Trials 2.1 Independent, Identically Distributed Trials In Chapter 1 we considered the operation of a CM. Many, but not all, CMs can be operated more than once. For example, a coin can be tossed or a die cast many times. By contrast, the next NFL season will operate only once. In this chapter we consider repeated operations of a CM. Let us return to the ‘Blood type’ CM of Section 1. Previously, I described the situation as follows: A man with AB blood and a woman with AB blood will have a child. The outcome is the child’s blood type. The sample space consists of A, B and AB. I stated in Chapter 1 that these three outcomes are not equally likely, but that the ELC is lurking in this problem. We get the ELC by viewing the problem somewhat differently, namely as two operations of a CM. The first operation is the selection of allele that Dad gives to the child. The second operation is the selection of allele that Mom gives to the child. The possible outcomes are A and B and it seems reasonable to assume that these are equally likely. Consider the following display of the possibilities for the child’s blood type. Allele from Mom Allele from Dad A B A A AB B AB B I am willing to make the following assumption. The allele contributed by Dad (Mom) has no influence on the allele contributed by Mom (Dad). Based on this assumption, and the earlier assumption of the ELC for each operation of the CM, I conclude that the four entries in the cells of the table above are equally likely. As a result, we have the following probabilities for the blood type of the child: P ( A ) = P ( B ) = 0 . 25 and P ( AB ) = 0 . 50 . Here is another example. I cast a die twice and I am willing to make the following assumptions. 11
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The number obtained on the first cast is equally likely to be 1, 2, 3, 4, 5 or 6. The number obtained on the second cast is equally likely to be 1, 2, 3, 4, 5 or 6. The number obtained on the first (second) cast has no influence on the number obtained on the second (first) cast. The 36 possible ordered results of the two casts are displayed below, where, for example, (5 , 3) means that the first die landed 5 and the second die landed 3. This is different from (3 , 5) . Number from Number from second cast first cast 1 2 3 4 5 6 1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Just like in the blood type example, b/c of my assumptions, I conclude that these 36 possibilities are equally likely. We will do a number of calculations now. For ease of presentation, define X 1 to be the number obtained on the first cast of the die and let X 2 denote the number obtained on the second cast of the die. We call X 1 and X 2 random variables , which means that to each possible outcome of the CM they assign a number. Every random variable has a probability distribution which is simply a listing of its possible values along with the probability of each value. Note that X 1 and X 2 have the same probability distribution; a fact we describe by saying that they are identically distributed ,
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