Math 3302_journal

Class Journal - Math 3302 (30094)

Summer I 2009

What I hear, I forget; what I see, I remember; what I do, I understand. – Kung Fu Tzu (Confucius)

One learns the thing by doing the thing; for though you think you know it,

You have no certainty until you try. – Sophocles

Session	Date	Read & Study Section	Discussion Topics	Practice Exercises	Other Info
20	7 - 2		Final Exam (Comprehensive) At usual class time (2:45 – 4:45) in usual class room A615. Bring a pencil and a scientific calculator (no cell phone calculators).
19	7 - 1	12.1&12.2	More details of the simple linear regression model were discussed, including SST, SSR and SSE. See pages 462-463.	See below.
18	6 - 30	12.1&12.2	Quiz 6 will be to use EXCEL to help solve a linear regression problem. Section 8.1&8.2 The least squares line is used to make predictions: (1) when you substitute an x-value into the equation of the least squares line, the matching y is the mean or expected value of y for that x – see page 451; (2) beware the danger of extrapolation – the trend of the data (linear relationship) may not hold for x-values that lie beyond the range of x-values in the data. See page 458. Also, the slope of the line gives the change in y when x increases by 1 unit. To measure how good the linear regression model is, a variance approach is used. The variance of the error term is denoted σ²: a large value of σ² indicates the observed data are quite spread out about the regression line, whereas, a small value of σ² indicates the observed data fall very close to the regression line. An estimate of σ² equals “the sum of squared errors divided by the number of ordered pairs less 2”=SSE/(n-2). See page 460. This variance idea is extended to get the coefficient of determination, which is denoted r². This can be interpreted as: r² is the proportion (percent) of observed variation that can be explained by the simple linear regression model (by the linear relationship) between x and y. See page 463. Intuitively, r² is a measure of how good the linear model fits the observed data. If the regression line passes exactly through every point on the scatter plot, then r² =1 and the model explains all of the variation in the y-values. The further the line is away from the data points, the less the model is able to explain the variation in the y-values and the closer r² is to 0. Note: 0<r²<1 for all data sets, and r² is a pure number with no units.	See below.
17	6 - 29	8.1&8.2 12.1&12.2	Section 8.1&8.2 When writing a conclusion in context of the problem, typically, you should refer to the alternative hypothesis and answer the question posed in the problem. The conclusion to example 8.6 is: “The data does not give strong support to the claim that the true average differs from the design value of 130” which is the alternative hypothesis. Exercise 28 was discussed. Section 8.1&8.2 Here, we generalize the deterministic linear model y=β₀+β₁x to a probabilistic model y=β₀+β₁x+ε , where the value of y cannot be determined exactly just from knowledge of x. Since we will use the equation of the line to make predictions, we call it a predicted y-value when we calculate y from the line. Also, we use β’s for the population model and b’s for the sample model. The criteria for the best fitting line is the line that has the smallest sum of squared errors Σ(observed y – predicted y)² from all possible lines drawn through the data; this line is also called the least squares line or regression line. See figure 12.3 on page 450. Calculus is used to find the coefficients of the least squares line: the formulas for the slope and y-intercept are on page 456. It is suggested that you use a column format to do these calculations by hand as shown in example 12.4 on page 457. A scatter plot of the sample is always a good first step in determining whether the data has a linear trend, or to determine if a linear model is appropriate. See pages 448-449. Please read and study sections 12.1 and 12.2. Note that all the calculations for the least squares line are available in software with statistical functions such as EXCEL; this was demonstrated in class.	See below and Section 12.2 17, 18, 19, 24 *Use EXCEL	Here are instructions on how to construct a scatter plot, add the least squares line, and compute the coefficient of determination in EXCEL 2007. If the data analysis button does not appear on your toolbar, you must install the "Analysis ToolPak." Here is a link to instructions on how to do this from Microsoft. http://office.microsoft.com/en-us/excel/HP100215691033.aspx Here is an interesting applet that lets you see how the least squares line minimizes the sum of squared errors: Simple regression applet
16	6 - 25		Test 2 (Sections 3.1-3.4, 4.1-4.3, 5.4, 7.1-7.2) Bring a pencil and a scientific calculator (no cell phone calculators). It is strongly recommended that you complete as many of the WebAssign homework problems as possible to help you prepare for the test.
15	6 – 24	8.1&8.2	Section 8.1&8.2 Some of the key ideas in a hypothesis test for a population mean µ are: (1) the value of µ is unknown and will remain unknown; (2) there are two competing and contradictory claims about µ that are called the null hypothesis and the alternative hypothesis; (3) the null hypothesis is assumed true and is not rejected unless the sample evidence significantly contradicts it; (4) a formula called the test statistic, whose distribution is known to be z or t, is chosen and into which the sample evidence will be substituted; (5) for each test, there is a significance level α; (6) the form of the alternative hypothesis together with the significance level α are used to determine the rejection region for the test; (7) there only two possible test results: reject the null hypothesis or do not reject the null hypothesis. Our text recommends a 7-step test of hypothesis procedure. See page 296. The rules for setting up the rejection region are given on page 296. No matter the test result, there is a chance that an error has been made because of an unrepresentative sample (bad data). If a true null hypothesis is rejected, this is called a Type I Error; if a false null hypothesis is not rejected, this is called a Type II Error. The notations used for the probabilities of the errors are: P(type I error)=α, the significance level, and P(type II error)=β. It is not possible to minimize both errors at the same time since decreasing the size of α increases the size of β, and vice versa. See page 292. Carefully study the detailed solutions given in example 8.6 on page 297 and example 8.8 on page 299.	See below.	Please note that for your convenience, the posted practice exercises in section 8.2 have attached a template for the 7 steps in a hypothesis test. There will not be a WebAssign HW assignment for Sections 8.1 & 8.2.
14	6 – 23	7.1&7.2 7.3 8.1&8.2	Sections 7.1&7.2 By the CLT, the z confidence interval formula for µ still gives good estimates if the sample size is large, n>30, and the sample standard deviation s is substituted for σ because it is unknown. In this case, the formula is and it is on page 264. This formula is used in example 7.6. Section 7.3 The random variable has a t distribution when sampling from a (1) normal population with (2) an unknown standard deviation σ and (3) the sample size is small, n<30. The properties of t distributions are: (1) each t curve is bell-shaped and centered at 0; (2) the spread of the curve depends on its degrees of freedom (df) and it is more spread out than the standard normal curve; (3) as the df becomes larger and larger, the t curves approach the standard normal curve. So the z curve is often called the t curve with df = . See page 271. Table A.5 in the appendix and in the inside of the back cover of the book gives Critical Values for t Distributions. See pages 271-272 for examples using this table. The small sample CI formula for µ based on a sample from a normal distribution with unknown σ is where df=n-1. This formula is used in example 7.11 on page 273. Section 8.1&8.2 In a hypothesis-testing problem, there are two competing claims about a population parameter like µ: the null hypothesis and the alternative hypothesis. A hypothesis test is a formal statistical procedure to determine if sample evidence contradicts the null hypothesis and provides strong support for the alternative hypothesis.	See below and Section 8.2 26, 28, 30
13	6 - 22	5.4 7.1&7.2	Class will begin with Quiz 5 (sections 3.4, 4.1 & 4.2). Section 5.4 In summary, to describe the distribution of the random variable X-bar means to describe the distribution of the means of all possible samples chosen from the same population and of the same size n. Theory says (a) the distribution of X-bar is centered at µ, the mean of the population being sampled; (b) the variance (spread) of X-bar equals σ²/n, the variance of the population being sampled divided by the sample size; (c) the shape of X-bar is a normal distribution for any size n if the population being sampled is a normal distribution, or the shape is a normal distribution for any population if n is sufficiently large (by Central Limit Theorem – the most important theorem in probability). Exercise 51 was discussed. Sections 7.1&7.2 Problem: Use a random sample to estimate an unknown population mean µ. Solution 1: Use the sample mean (a point estimate) to estimate µ. Disadvantages: This answer gives no information about the reliability or precision of the estimate. Solution 2: Use a confidence interval estimate to estimate µ. Advantages: 1. The confidence level gives information about the reliability of the procedure. For example, a confidence level of 95% implies that 95% of all possible samples would give an interval that includes µ, and only 5% would yield an erroneous interval that does not include µ. 2. The width of the interval conveys the precision of the estimate. The smaller the width of the interval, the better the precision. Note: the margin of error E=2(width). The CI formula is used when sampling from (1) a normal population with (2) known standard deviation σ and (3) any size n. The z critical value depends on the confidence level. See page 258. To ensure an interval that has a specified confidence level and specified width, the necessary sample size n can be determined. See the formula on page 260. Be certain to round up when using this formula. Exercise 5 was discussed.	See below and Section 7.3 28, 30, 32, 33c, 35a
12	6 - 18	4.3 5.4	Section 4.3 In future statistical inference problems, we will need the value of z that captures a right tail of area α for the standard normal curve. This z-value is denoted by the notation z_α and it is also called a z critical value. For example z_0.05 = 1.645 means that z=1.645 cuts off a right tail of area 0.05. This also means that z=1.645 is the 95^th percentile of the standard normal distribution. See pages 148 and 149 on how to find z critical values. The standardization formula Z=(X-µ)/σ can be rewritten as X= µ+Zσ. This latter formula is useful when finding percentiles for a random variable X with a normal distribution. The strategy is to first find the percentile for Z, and then second substitute into the second formula to find the matching X. See example 4.18 on page 151. Section 5.4 Suppose you have a population with mean µ and standard deviation σ. Intuitively, before you choose a sample, the sample mean X-bar for a sample of size n from this population cannot be predicted in advance; it most likely will not equal µ and most likely will change from sample to sample. Therefore, the sample mean X-bar is a random variable. There are theory rules that tell us the mean, variance and shape of the distribution of X-bar: (a) the mean of X-bar equals µ, the mean of the population being sampled; (b) the variance of X-bar equals σ²/n, the variance of the population being sampled divided by the sample size; (c) the shape is a normal distribution for any size n if the population being sampled is a normal distribution, or the shape is a normal distribution for any population if n is sufficiently large. The later statement is called the Central Limit Theorem (CLT). The rule of thumb is that the CLT applies if n>30. For this reason, a large sample is one with sample size n>30. See pages 213-215. These rules help us determine properties of X-bar before a sample is chosen. See example 5.26 on page 215. Exercises 46 and 50 were discussed.	See below and Section 7.1 & 7.2 4, 5, 12, 13, 14
11	6 - 17	4.1 4.2 4.3	Class will begin with Quiz 4 (sections 3.1, 3.2 & 3.3). Section 4.1 Note the discussion on page 133 on why P(X=c)=0 for a continuous random variable, and on the consequence that the probability X lies in an interval between a and b does not depend on whether a or b is included. Section 4.2 Note on page 136 when computing the cumulative distribution function F(x) that the integrand is f(y); f(t) can also be used. The cdf F(x) can be used to compute probabilities of intervals: P(X>a)=1-F(a); P(a<X<b)=F(b)-F(a). This is on page 137. The pdf f(x) can be obtained from the cdf F(x) by differentiation: F’(x)=f(x). See page 139. Section 4.3 To say a continuous rv X has a normal distribution means that the pdf f(x) of X has the complex formula shown on page 145. The parameters µ and σ in the formula are the mean and standard deviation, respectively, of X. Also, the graph of f(x) is a symmetric, bell-shaped curve that is centered at µ and has inflection points at a distance of σ from µ. See figure 4.13 on page 145. The normal distribution with µ=0 and σ=1 is called the standard normal distribution, and this random variable is denoted by Z. Also, the cdf of Z is denoted by Φ(z). Appendix Table A.3 gives values of Φ(z), which correspond to the area to the left of z under the standard normal curve. See page 146. Study example 4.13 to see how to calculate probabilities for the standard normal variable Z. If X is normally distributed with mean µ and standard deviation σ, then to find probabilities involving X, we standardize by making a change of variable from X to Z. See the formula on page 149. Study examples 4.16 and 4.17 on pages 150-151 to see how to standardize from X to Z. When we say a person’s test score is at the 85^th percentile, we mean that 85% of all scores are below that score and that 15% are above. Similarly, to find the 85^th percentile of the standard normal distribution, means to find the z-score so that the area to the left of that score is 0.8500, that is, so that Φ(z)=.8500. To solve this, we find the closest value in the body of the table to .8500, above or below, and then find the matching z-score. Note that we average two z-scores when the percentile is the midpoint between two values in the body of the table. Study example 4.14 on page 147.	See below and Section 5.4 46, 47, 48, 50, 51, 53
10	6 - 16	3.4 4.1 4.2	Section 4.1 The probability distribution function or probability density function (pdf) f(x) of a continuous rv X has 3 properties: (1) f(x)>0 for all x; (2) the area under f(x) from negative infinity to positive infinity equals 1; (3) P(a<X<b)=”area under f(x) between x=a and x=b.” See figure 4.2 on page 132. Section 4.2 The formulas for the mean, variance and cumulative distribution of a continuous rv have a similar pattern to those for a discrete rv, except the summation sign Σ is replaced with an integral sign , and p(x) is replaced with f(x). See pages 136, 141 and 142. Section 4.1 The probability distribution function or probability density function (pdf) f(x) of a continuous rv X has 3 properties: (1) f(x)>0 for all x; (2) the area under f(x) from negative infinity to positive infinity equals 1; (3) P(a<X<b)=”area under f(x) between x=a and x=b.” See figure 4.2 on page 132. Section 4.2 The formulas for the mean, variance and cumulative distribution of a continuous rv have a similar pattern to those for a discrete rv, except the summation sign Σ is replaced with an integral sign , and p(x) is replaced with f(x). See pages 136, 141 and 142. Section 3.4 A binomial experiment consists of n trials; there are only 2 outcomes S and F on each trial; p=P(S) and (1-p)=P(F) is the same on each trial; the random variable X is the number of successes in the n trials; P(x successes and n-x failures)=C_x,n ×p^x ×(1-p)^n-2 . This is the formula for the binomial pmf and is abbreviated b(x;n,p) as shown on page 111. The cumulative binomial probability function is abbreviated B(x;n,p) as shown on page 112. Table A.1 in the appendix gives cumulative binomial probability values for n=5, 10, 15, 20, 25 in combination with selected values of p. There are shortcut formulas for the mean and variance of a binomial rv on page 113: E(X)=np and V(X)=np(1-p). Exercise 54 was discussed. Section 4.1 The probability distribution function or probability density function (pdf) f(x) of a continuous rv X has 3 properties: (1) f(x)>0 for all x; (2) the area under f(x) from negative infinity to positive infinity equals 1; (3) P(a<X<b)=”area under f(x) between x=a and x=b.” See figure 4.2 on page 132. Section 4.2 The formulas for the mean, variance and cumulative distribution of a continuous rv have a similar pattern to those for a discrete rv, except the summation sign Σ is replaced with an integral sign , and p(x) is replaced with f(x). See pages 136, 141 and 142.	See below and Section 4.1 & 4.2 1, 3, 5, 12, 15, 19 Section 4.3 28, 29, 31, 32, 33, 34, 38, 39, 40, 44, 47	You can download the Appendix Tables from the text website.
9	6 - 15	3.2 3.3 3.4	Section 3.2 There was a review of properties of the probability distribution of a discrete random variable. Problems 12 and 23 were discussed. Section 3.3 The expected value of a discrete random variable X is also the called the mean value of X and is denoted by µ_X or sometimes just by µ when it is clear to which X the expected value refers to. In theory, the expected value is the mean of the infinite population that you get when the experiment is repeated over and over – the average in the long run. Note in example 3.18, that the mean of the binomial rv with values 1 and 0, and probabilities p and (1-p), respectively, is E(X)=p. The expected value of a function of X, h(X), is defined on page 103: just substitute h(x) in place of x in the expected value formula for E(X). Sometimes the formula for h(X) is given in a problem, and sometimes you must set up the formula (see examples 3.21 and 3.23 on page 103). To find the expected value of a linear function of X, h(X)=aX+b, just use E(X) as the input to h(X), that is, E[h(X)]= h[E(X)]=aE(X)+b. See page 104. The variance of X is the expected value of the function h(X)=(X- µ)², and the standard deviation of X is the square root of the variance, see page 105. There is also a shortcut formula for the variance: V(X)=E(X²)-[E(X)]². And, there are rules for the variance of a linear function h(X)=aX+b on page 106. In a game of chance, if X is the possible winnings and p(X) is the matching probability, then E(X) gives the average winnings per game, in the long run. Examples of expected value in lottery games and in insurance were shown. Section 3.4 Every binomial experiment has 4 common properties – see page 108. The classic example of a binomial experiment is to flip a coin a definite number of times. In a binomial experiment of n trials with each trial having only the two possible outcomes S and F, the random variable X equals the number of S’s in the n trials. The pmf for X is on page 111.	See below and Section 3.4 46, 47, 48, 50, 54, 55, 60, 65	Try your luck: empirically determine the expected winnings of a game at http://www.stat.tamu.edu/~west/applets/expvalue.html Here is a link to some sample actuarial exams http://www.beanactuary.org/exams/exam_sample.cfm
8	6 - 11		Test 1 (Sections 2.1, 2.2, 2.3, 2.4, 2.5, 1.4) Bring a pencil and a scientific calculator (no cell phone calculators). It is strongly recommended that you complete as many of the WebAssign homework problems as possible to help you prepare for the test.
7	6 - 10	3.2 3.3	Class will begin with Quiz 3 (sections 2.4 & 2.5). Section 3.2 A probability distribution p(x) of a discrete random variable is also called a probability mass function (pmf) – see why on page 93. From the pmf p(x), a new function F(x) can be created that is called the cumulative distribution function (cdf). For any number x, F(x) equals the probability that X is at most (<) x. So F(2)=probability X is at most 2=p(0)+p(1)+p(2). See page 95. Properties of the cdf: (1) P(a<X<b)=F(b)-F(a-1); (2) P(X=a)=F(a)-F(a-1); when a and b are integers. Section 3.3 The mean or expected value of a discrete random variable X is a weighted average of the possible values of X where the weights are the probabilities of those values. See page 101. Note that the mean is average value of X, in the long run. See page 102.	See below and Section 3.3 29, 30, 32, 39
6	6 - 9	2.3 1.4 3.1 3.2	Class will begin with Quiz 2 (sections 2.2 & 2.3). Section 2.3 Carefully study example 2.23 on page 64 “randomly select 6 computers.” The solution method can be applied to many other problems. Section 1.4 Note that if the variance of a data set equals zero, then all of the data have the same value – there is no variability since all the values are equal. Section 3.1 For a given sample space of some experiment, a random variable is a rule (function) that assigns a number to each possible outcome. See page 87. A Bernoulli random variable only has possible values of 0 and 1. A random variable is classified as discrete or continuous, depending on the number of possible values that it may have: discrete or continuous. See page 89. Section 3.2 The probability distribution of a discrete random variable gives the possible values of the random variable and the probability of each. Every probability distribution must satisfy 2 requirements: (1) p(x)>0 for each x; (2) Σp(x)=1 where the sum is over all possible values of x. See page 91. A probability distribution may have different forms: a table, a line graph, a histogram, or a formula.	See below and Section 3.2 11, 12, 13, 15, 16, 23
5	6 - 8	2.5 1.4	Section 2.5 Let A and B be events; each with nonzero probability. By definition, the events A and B are mutually exclusive if “A intersect B” equals the null event φ; that is, A and B cannot occur at the same time; And, A and B are independent if P(A\|B)=P(A). Now put these two ideas together. For mutually exclusive events A and B: if B occurs, then A cannot occur, since P(A intersect B) = 0. Hence, P(A\|B)=P(A intersect B)/P(B)=0 which does not equal P(A), a nonzero number. Therefore, A and B are not independent; they are dependent. See example 2.32 on page 77. Exercise 71 was discussed. Section 1.4 A number that locates the center of a data set is called a measure of location. The median and mean are both measures of location. Usually, the mean is the best measure of location, but if there is an outlier in the data set, then it may be better to use the median. The median is the middle value of the ordered data when n is odd; the median is the average of the two middle values of the ordered data when n is even. See page 27. Measures of variability give information about the spread of the data values. The simplest such measure is the range, which equals the highest value minus the lowest value in the data set, or H – L. The most important measure of variability is the standard deviation (of the mean), which equals the square root of the variance. The sample variance s² equals the sum of the squared deviations from mean divided the number of values less one; see page 32. An alternate “computational” formula for the variance is on page 34. Properties of the variance and standard deviation are on page 35: (a) adding (subtracting) a constant to each data value, changes the mean but not the variance/standard deviation; (b) multiplying (dividing) by a constant, changes the mean and the variance/standard deviation. An example was done in class on how to calculate the sample variance and sample standard deviation by hand, and by using EXCEL.	See below and Section 3.1 1, 6, 7, 8
4	6 - 4	2.4 2.5	Section 2.4 The law of total probability is used in Bayes’ theorem: if the sample space S=A₁UA₂ where A₁ and A₂are disjoint, then for any event B with nonzero probability, the posterior probability of A₁ given that B has occurred is P(A₁\|B)=P(A₁∩B)/P(B) = P(A₁∩B) / [P(B∩A₁)+P(B∩A₂)]. This rule extends to more than two A_i’s. See page 73. Study example 2.30, incidence of a rare disease, to see how Bayes’ theorem is applied. Pay attention to the tree diagram in the solution to this example – it is very helpful. Exercise 63 was discussed. Section 2.5 In mathematics, the following statements are equivalent: (1) two events A and B are independent; (2) P(A\|B)=P(A); (3) P(B\|A)=P(B); (4) P(A∩B)= P(A)∙P(B), this is a special case of the product rule for P(A∩B). Also, if A and B are independent events, then so are the 3 pairs A' and B, A and B', and A' and B'. See page 77. More than two events are said to be mutually independent if the probability of the intersection of any subset of the events is equal to the product of the individual probabilities. See page 79. Exercises 77 and 80 were discussed.	See below and Section 1.4 45, 46, 47, 49, 51	Here is the link to an article on Bayes’ theorem that you may find of interest: The Legacy of Reverend Bayes.
3	6 - 3	2.3 2.4	Class will begin with Quiz 1 (section 2.1). Section 2.3 Some applications of counting rules: Counting rules may be applied in probability problems that have equally likely outcomes since P(A)=(# outcomes in A)/(# outcomes in S). In example 2.23, 6 printers are to be chosen randomly from 25 printers, of which 10 are laser and 15 are inkjet. Then the number of ways to choose 3 laser and 3 inkjet printers is C_3,15 ×C_3,10 and the number of ways to choose 6 printers is C_6,25 . Therefore, the probability of choosing 3 laser and 3 inkjet printers equals (C_3,15 ×C_3,10)/ C_6,25. See pages 64-65. Exercises 29, 38 and 42 were discussed. Section 2.4 Intuitively, the conditional probability of A given that the event B has occurred means that we want to find the probability of A in a smaller sample space – we will only consider outcomes favorable to B. The definition of P(A\|B), the conditional probability of A given that B has occurred, is given on page 68. This is also referred to as “probability of A if B has occurred.” From the formula for conditional probability, a new formula can be derived that is called the multiplication rule for P(A∩B), the probability of A and B: P(A∩B)=P(A)∙P(B\|A)=P(B)∙P(A\|B). See page 69. Study example 2.27 to see how to apply this multiplication rule. The Law of Total Probability: (1) if the sample space S=A₁UA₂ where A₁ and A₂are disjoint, then any event B is partitioned into B=(B∩A₁)U(B∩A₂) where (B∩A₁) and (B∩A₂) are disjoint, and so P(B)=P(B∩A₁)+P(B∩A₂); (2) this rule extends to any finite number of A_i’s. See page 72.	See below and Section 2.5 71, 73, 74, 77, 80, 82, 83, 84
2	6 - 2	2.2 2.3	Section 2.2 Propositions (Properties) of Probability that can be proven: (1) Probability of null event P(φ)=0, page 52; (2) Complement rule P(A)+P(A')=1, page 54; (3) P(A)<1, page 55, which with axiom 1 gives range rule 0<P(A)<1; (4) General sum rule P(AUB)=P(A)+P(B)-P(A∩B), page 55; (5) Union of three events rule P(AUBUC)=P(A)+P(B)+P(C)-P(A∩B)-P(A∩C)-P(B∩C)+P(A∩B∩C), page 56; (6) Equally likely outcomes rule when there are N equally likely outcomes then P(A)=(number of outcomes in A)/(number of possible outcomes)=N(A)/N, page 57. It is useful to draw a Venn diagram in problems – see page55 and 56. Exercises 12 and 19 were discussed. Section 2.3 One version of the Product Rule is if two operations are to be performed one after the other and the first operation can be done in n₁ ways and for each of these the second operation can be done in n₂ ways, then both operations can be done in n₁n₂ ways. A second version is on page 60. The General Product Rule extends the product rule to more than two operations (to k-tuples); see page 61. The product rule can be illustrated by a figure called a tree diagram; see pages 60-61. Factorial notation is useful in many counting problems; see page 63. An ordered arrangement of distinct objects is called a permutation. The formula for the number of permutations (ordered arrangements) of size k chosen from n objects is given on page 63. A subset (order is not important) of a collection of objects is called a combination. The formula for the number of combinations (unordered subsets) of size k chosen from n objects is given on page 64. Study the examples in section 2.3.	See below and Section 2.3 29, 31, 32, 35, 38, 42, 43 Section 2.4 45, 46, 47, 50, 51, 55, 59, 60, 62, 63, 64	Here is a link to a web site with more information on blood types related to problem 45, section 2.4: http://www.givelife2.org/aboutblood/bloodtypes.asp
1	6 - 1	2.1 2.2	Section 2.1 Carefully study the definitions in this section: experiment, sample space, event (simple or compound), complement A', union AUB, intersection A∩B, null event φ, mutually exclusive or disjoint events, Venn diagram. It takes practice to be able to translate a verbal description of an event into symbols with the operations of union, intersection, and complementation, and also to be able to draw an appropriate Venn diagram. See exercise 8, page 51. Exercises 1, 3, 7, 8 and 9 were discussed in class. Section 2.2 For any event A, the probability of A, which is denoted P(A) equals a number which measures the chance that A will occur. There are three axioms (basic properties) of probability given on page 51: (i) the nonnegative axiom; (ii) the sample space S has probability 1; (iii) the sum rule for a countably infinite collection of disjoint (mutually exclusive) events. The relative frequency interpretation of probability: if the experiment is repeated a large number of times, the probability of an event gives the proportion (percent) of the time that the event occurs. See pages 53 and 54.	Section 2.1 1, 3, 7, 8, 9 Section 2.2 11, 12, 13, 14, 15, 19, 21, 22	Please register in WebAssign as soon as possible; instructions are given on the syllabus.