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Prof. Thistleton STA100 Statistical Methods Lecture 6
Text Sections: Chapter 4 Sections 2
Working With Sets
Recall the table from Lecture 5.
Gender Male Female
Owns a Pet
Yes 15 27 No 6 3
We can think about this in a couple of ways. There are 51 students in this sample drawn from the population of SUNYIT students. With each of these students we have associated two variables: Gender and Owns a Pet. Both of these variables are categorical, as we discussed in the first lecture. You could also say that each is dichotomous. All we mean by that is that each one can occur in two ways: Yes or No. Later weโll call that Success or Failure.
We could also think a little differently with each individual having membership in Male or Female and in Owns a Pet or Does Not Own A Pet. You might recall from earlier math courses that we can begin to talk about membership in a set. A picture, called a Venn Diagram, often helps.
Before proceeding further we need to develop a few ideas about collections of objects, or sets.
Male
Owns
a Pet
- Definition 1 A set is a collection of objects. These objects are called elements of the set. We typically will use an upper case letter, such as A, for a set and a lower case letter, such as a, for an element of a set. We write a โ A with the Greek letter โlittle epsilonโ.
Example: If Mike owns a pet we would say that ๐๐๐๐ โ ๐๐๐๐ ๐ด ๐๐ธ๐. Note that we would also say that ๐๐๐๐ โ ๐๐ด๐ฟ๐ธ.
- Definition 2 A subset, say B, of a set A, is itself a set, every element of which also belongs to the set A. We write ๐ต โ ๐ด. That is, ๐ต is just a collection of some (and maybe all) of the elements of A.
Example: the New York Yankees 2008 40 man roster http://newyork.yankees.mlb.com/team/roster_40man.jsp?c_id=nyy includes subsets corresponding to pitchers, catchers, infielders, outfielders, and designated hitters. Note that these subsets donโt have any overlap- no one is both a pitcher and an infielder.
3. Definition 3 Two sets, A and B, are said to be equal if each element of A is also in B, and if each element of B is also in A. We write B = A. 4. Definition 4 We will call the set C the union of the sets A and B if every element of C is also in at least one of A or B (though maybe in both). We write ๐ถ = ๐ด ๐๐ ๐ต. Note that other books will write ๐ถ = ๐ด โช ๐ต.
Example: Suppose we toss a fair coin 3 times and observe the faces. What does our sample space look like? We can list these outcomes out in a straightforward way by noting that on one toss the sample space looks like ๐ป, ๐. (Note: exactly 2 outcomes). Build up from here by noting that if we toss twice then the sample space looks like ๐ป๐ป, ๐ป๐, ๐๐ป, ๐๐. (Note: exactly 4 outcomes. To help see this, draw a โtree diagramโ as follows
outcomes. For the first subset, call it ๐ธ 1 we can list the outcomes with โHeads on Firstโ. Can you see that ๐ธ 1 = ๐ป๐ป๐ป, ๐ป๐ป๐, ๐ป๐๐ป, ๐ป๐๐? Now let the second event, call it ๐ธ 2 be those outcomes with โHeads on Secondโ. We can write this out as ๐ธ 2 = ๐ป๐ป๐ป, ๐ป๐ป๐, ๐๐ป๐ป, ๐๐ป๐. Hereโs something to think about: The event ๐ธ 1 has 4 outcomes. The event ๐ธ 2 has 4 outcomes. How many does ๐ธ 1 ๐๐ ๐ธ 2 have? Itโs not 8- just write them out, being careful not to write any outcomes out twice: ๐ธ 1 ๐๐ ๐ธ 2 = ๐ป๐ป๐ป, ๐ป๐ป๐, ๐ป๐๐ป, ๐ป๐๐, ๐๐ป๐ป, ๐๐ป๐ This is a terrifically important result for us. To state it best we also need the following.
5. Definition 5 We will call the set C the intersection of the sets A and B if every element of C is also in both of ๐ด ๐๐๐ ๐ต. We write ๐ถ = ๐ด ๐๐๐ ๐ต. Note that other books will write ๐ถ = ๐ด โฉ ๐ต.
Example: still working with the above, we have ๐ธ 1 ๐๐๐ ๐ธ 2 = ๐ป๐ป๐ป, ๐ป๐ป๐. That is, we include all those tosses with Heads on First and also Heads on Second. This allows us to say that โthe number of elements in A or B is equal to the number of elements in A plus the number of elements in B minus the number of elements in both. Using the symbol ๐ ๐ด for โthe number of elements in Aโ we can write this succinctly as
๐ ๐ด ๐๐ ๐ต = ๐ ๐ด + ๐ ๐ต โ ๐ ๐ด ๐๐๐ ๐ต
In our coin toss example this gives us
๐ ๐ป ๐๐ 1 ๐ ๐ก ๐๐ ๐ป ๐๐ 2 ๐๐ = ๐ ๐ป ๐๐ 1 ๐ ๐ก + ๐ ๐ป ๐๐ 2 ๐๐ โ ๐ ๐ป ๐๐ 1 ๐ ๐ก ๐๐๐ ๐ป ๐๐ 2 ๐๐
In our Pets and Gender example this gives us:
Gender Male Female
Owns a Pet
Yes 15 27 42 No 6 3 9 21 30 51
And so we see that ๐ ๐๐ค๐๐ ๐ ๐๐๐ก = 42 and ๐ ๐๐๐๐ = 21_. We also have that_ ๐ ๐๐๐๐ ๐๐๐ ๐๐ค๐๐ ๐ ๐๐๐ก = 15 so that, finally,
๐ ๐๐ค๐๐ ๐ ๐๐๐ก ๐๐ ๐๐๐๐ = 42 + 21 โ 15 = 48
Bringing this back home to probability (remember Kolmogorovโs third axiom!!)
๐ ๐๐ค๐๐ ๐ ๐๐๐ก ๐๐ ๐๐๐๐ =^4251 +^2151 โ 1551 = (^4851)
6. Definition 6 Suppose we have ๐ต โ ๐. Then we denote the complement of the set B in ๐ as the set of elements which are in ๐ but not in ๐ต. That is, just those elements not in ๐ต. We write ๐๐๐ก ๐ต.
In particular, consider the set of all outcomes of an experiment. We will call this set the sample space of the experiment and denote it as S. An event is then just a subset of a sample space and we may speak of an event ๐ธ and its complement ๐๐๐ก ๐ธ and their probabilities.
We have a few relationships to help us do our work:
- ๐ด ๐๐
๐๐๐ก ๐ด = ๐. That is, an event and its complement together make up the sample space.
- ๐๐๐ก ๐ด ๐ด๐๐ท ๐ต = ๐๐๐ก ๐ด ๐๐
๐๐๐ก ๐ต. That is, the complement of the intersection is the union of the complements.
- ๐๐๐ก ๐ด ๐๐
๐ต = ๐๐๐ก ๐ด ๐ด๐๐ท ๐๐๐ก ๐ต. That is, the complement of the union is the intersection of the complements.
๐๐๐๐ ๐ธ ๐๐
๐น = ๐๐๐๐ ๐ธ + ๐๐๐๐ ๐น โ ๐๐๐๐ ๐ธ ๐ด๐๐ท ๐น and so ๐๐๐๐ ๐ธ ๐๐
๐น = 0.5 + 0.2 โ 0.15 = 0.
- Calculate the probability that you do not stop. This is just the complement of the above and so has probability 1 โ 0.55 = 0.
- Calculate the probability that you stop exactly once. To answer this we can revisit our diagram and embellish it a little. Since ๐ธ has two parts which together give us a probability of 0.5, and since the part taken up by the intersection is of size 0.15, that leaves us 0.35 left over for the event โstop at first but not at secondโ.
Similarly, the probability that we stop at the second but not the first is 0.
- 0.15 = 0.05. So, the probability we stop just once is 0.35 + 0.05 = 0.4.
0.35 0.1^5
0.
0.4 5
Example: You will roll a fair die three times.
- How many possible sequences (outcomes) are there?
- What is the probability of that you will obtain 0 twos?
- What is the probability of that you will obtain 1 two?
- What is the probability of that you will obtain 2 twos?
- What is the probability of that you will obtain 3 twos?
- What is the probability of that, if you add the face values, you will obtain a sum of 17 or more?
