Chapter 12 EPS Due 11/20/2015 50 points

A _____________ phenomenon (chance experiment) has outcomes that we cannot Predict, but that nonetheless have a _____________ distribution in very many repetitions. The probability of an event is the _____________ of times the event occurs in many ________________ trials of a random phenomenon. A probability model for a random phenomenon consists of a _________ ____________ S and an assignment of ________________ P. The __________ ______________S is the set of all possible outcomes of the random phenomenon. Sets of outcomes are called __________. P assigns a number P(A) to an event A as its __________________ .

Any assignment of probability must obey the rules that state the basic properties of probability: _____ ≤ P(A) ≤ ________ for any event A. 2. P(S) = ________. 3. Addition rule for disjoint events: Events A and B are __________ if they have no outcomes in common. If A and B are disjoint, then P(A or B) = ________ + __________. 4. For any event A, P(A does not occur) = 1 – __________.

When a sample space S contains finitely many possible values, a finite __________ model assigns each of these values a probability between ____ and ______ such that the sum of all the probabilities is exactly __________. The probability of any event is the sum of the probabilities of all the values that make up the event. Finite probability models are also referred to as ___________ probability models. A sample space can contain all ___________ in some interval of numbers. A ___________ probability model assigns probabilities as _____________ under a density _____________. The probability of any event is the ____________ under the curve above the values that make up the event.

A random variable is a ____________ taking numerical values determined by the ____________ of a random phenomenon. The __________ ________________ of a random variable X tells us what the possible values of X are and how probabilities are assigned to those values. A random variable X and its distribution can be ____________ or _______________. The distribution of a ____________ random variable with finitely many possible values gives the ___________ of each value. A continuous_____________ random variable takes all values in some interval of numbers. A density _____________ describes the probability distribution of a continuous random variable.

Chapter Specifics A random phenomenon (chance experiment) has outcomes that we cannot predict but that nonetheless have a regular distribution in very many repetitions. The probability of an event is the proportion of times the event occurs in many repeated trials of a random phenomenon. A probability model for a random phenomenon consists of a sample space S and an assignment of probabilities P. The sample space S is the set of all possible outcomes of the random phenomenon. Sets of outcomes are called events. P assigns a number P(A) to an event A as its probability. Any assignment of probability must obey the rules that state the basic properties of probability: 1. 0 ≤ P(A) ≤ 1 for any event A. 2. P(S) = 1. 3. Addition rule for disjoint events: Events A and B are disjoint if they have no outcomes in common. If A and B are disjoint, then P(A or B) = P(A) + P(B). 4. For any event A, P(A does not occur) = 1 – P(A).

When a sample space S contains finitely many possible values, a finite probability model assigns each of these values a probability between 0 and 1 such that the sum of all the probabilities is exactly 1. The probability of any event is the sum of the probabilities of all the values that make up the event. Finite probability models are also referred to as discrete probability models. A sample space can contain all values in some interval of numbers. A continuous probability model assigns probabilities as areas under a density curve. The probability of any event is the area under the curve above the values that make up the event. A random variable is a variable taking numerical values determined by the outcome of a random phenomenon. The probability distribution of a random variable X tells us what the possible values of X are and how probabilities are assigned to those values. A random variable X and its distribution can be discrete or continuous. The distribution of a discrete random variable with finitely many possible values gives the probability of each value. A continuous random variable takes all values in some interval of numbers. A density curve describes the probability distribution of a continuous random variable.

12.22 You read in a book on poker that the probability of being dealt two pairs in a five-card poker hand is 1/21. This means that (a)if you deal thousands of poker hands, the fraction of them that contain two pairs will be very close to 1/21. (b)if you deal 21 poker hands, exactly one of them will contain two pairs. (c)if you deal 21,000 poker hands, exactly 1000 of them will contain two pairs. 12.23 A basketball player shoots five free throws during a game. The sample space for counting the number she makes is (a)S = any number between 0 and 1. (b)S = whole numbers 0 to 5. (c)S = all sequences of five hits or misses, like HMMHH.

Here is the probability model for the political affiliation of a randomly chosen adult in the United States.10 Exercises 12.24, 12.25, 12.26, and 12.27 use this information. 12.24 This probability model is (a)continuous. (b)finite. (c)equally likely. 12.25 The probability that a randomly chosen American adult’s political affiliation is “Other” must be (a)any number between 0 and 1. (b)0.03. (c)0.3. 12.26 What is the probability that a randomly chosen American adult is a member of one of the two major political parties (Republicans and Democrats)? (a)0.42 (b)0.55 (c)0.97 12.27 What is the probability that a randomly chosen American adult is not a Republican? (a)0.25 (b)0.75 (c)0.03

12.28 In a table of random digits such as Table B, each digit is equally likely to be any of 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. What is the probability that a digit in the table is a 7? (a)1/9 (b)1/10 (c)9/10 12.29 In a table of random digits such as Table B, each digit is equally likely to be any of 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. What is the probability that a digit in the table is 7 or greater? (a)7/10 (b)4/10 (c)3/10

12.30 Choose an American household at random, and let the random variable X be the number of cars (including SUVs and light trucks) the residents own. Here is the probability model if we ignore the few households that own more than six cars: A housing company builds houses with two-car garages. What percent of households have more cars than the garage can hold? (a)14% (b)22% (c)42% 12.31 Choose a common fruit fly Drosophila melanogaster at random. Call the length of the thorax (where the wings and legs attach) Y. The random variable Y has the Normal distribution with mean μ = 0.800 millimeter (mm) and standard deviation σ = 0.078 mm. The probability P(Y > 1) that the fly you choose has a thorax more than 1 mm long is about (a)0.995. (b)0.5. (c)0.005.

12.32 Sample space. In each of the following situations, describe a sample space S for the random phenomenon. (a)A basketball player shoots four free throws. You record the sequence of hits and misses. (b)A basketball player shoots four free throws. You record the number of baskets she makes. 2 points

12.33 Probability models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. Remember, a legitimate model need not be a practically reasonable model. If the assignment of probabilities is not legitimate, give specific reasons for your answer. (a)Roll a six-sided die, and record the count of spots on the up-face: P(1) = 0 P(2) = 1/6 P(3) = 1/3 P(4) = 1/3 P(5) = 1/6 P(6) = 0 (b)Deal a card from a shuffled deck: P(clubs) = 12/52 P(diamonds) = 12/52 P(hearts) = 12/52 P(spades) = 16/52 (c)Choose a college student at random and record sex and enrollment status: P(female full-time) = 0.56 P(male full-time) = 0.44 P(female part-time) = 0.24 P(male part-time) = 0.17 3 points

12.34 Education among young adults. Choose a young adult (aged 25 to 29) at random. The probability is 0.10 that the person chosen did not complete high school, 0.27 that the person has a high school diploma but no further education, and 0.34 that the person has at least a bachelor’s degree. (a)What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor’s degree? (b)What is the probability that a randomly chosen young adult has at least a high school education? 2 points

12.37 Car colors. Choose a new car or light truck at random and note its color. Here are the probabilities of the most popular colors for vehicles sold globally in 2013:12 (a)What is the probability that the vehicle you chose has any color other than those listed? (b)What is the probability that a randomly chosen vehicle is neither white nor silver?

12.38 Drawing cards. You are about to draw a card at random (that is, all choices have the same probability) from a set of seven cards. Although you can’t see the cards, here they are: (a)What is the probability that you draw a 9? (b)What is the probability that you draw a red 9? (c)What is the probability that you do not draw a 7?

12. 39 Loaded dice. There are many ways to produce crooked dice 12.39 Loaded dice. There are many ways to produce crooked dice. To load a die so that 6 comes up too often and 1 (which is opposite 6) comes up too seldom, add a bit of lead to the filling of the spot on the 1 face. If a die is loaded so that 6 comes up with probability 0.2 and the probabilities of the 2, 3, 4, and 5 faces are not affected, what is the assignment of probabilities to the six faces? 12.40 A door prize. A party host gives a door prize to one guest chosen at random. There are 48 men and 42 women at the party. What is the probability that the prize goes to a woman? Explain how you arrived at your answer.

Choose at random a person aged 15 to 44 years Choose at random a person aged 15 to 44 years. Ask their age and who they live with (alone, with spouse, with other persons). Here is the probability model for 12 possible answers 12.42 Living arrangements. (a)Why is this a legitimate finite probability model? (b)What is the probability that the person chosen is a 15- to 19-year-old who lives with others (not a spouse)? (c)What is the probability that the person is 15 to 19 years old? (d)What is the probability that the person chosen lives with others (not a spouse)? 3 points

12.43 Living arrangements, continued. (a)List the outcomes that make up the event A ={the person chosen is either 15 to 19 years old or lives with others, or both} (b)What is P(A)? Explain carefully why P(A) is not the sum of the probabilities you found in parts (c) and (d) of Exercise 12.42. 12.44 Living arrangements, continued. (a)What is the probability that the person chosen is 20 years old or older? (b)What is the probability that the person chosen does not live alone? 2 points 2 points

12.48 Birth order. A couple plans to have three children. There are eight possible arrangements of girls and boys. For example, GGB means the first two children are girls and the third child is a boy. All eight arrangements are (approximately) equally likely. (a)Write down all eight arrangements of the sexes of three children. What is the probability of any one of these arrangements? (b)Let X be the number of girls the couple has. What is the probability that X = 2? (c)Starting from your work in part (a), find the distribution of X. That is, what values can X take, and what are the probabilities for each value? 4 points

12.51 Random numbers. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose you specify that the random number Y can take any value between 0 and 2. Then the density curve of the outcomes has constant height between 0 and 2, and height 0 elsewhere. (a)Is the random variable Y discrete or continuous? Why? (b)What is the height of the density curve between 0 and 2? Draw a graph of the density curve. (c)Use your graph from part (b) and the fact that probability is area under the curve to find P(Y ≤ 1).

12.52More random numbers. Find these probabilities as areas under the density curve you sketched in Exercise 12.51. (a)P(0.5 < Y < 1.3) (b)P(Y ≥ 0.8)

12.53 Survey accuracy. A sample survey contacted an SRS of 2854 registered voters shortly before the 2012 presidential election and asked respondents whom they planned to vote for. Election results show that 51% of registered voters voted for Barack Obama. We will see later that in this situation the proportion of the sample who planned to vote for Barack Obama (call this proportion V) has approximately the Normal distribution with mean μ = 0.51 and standard deviation σ = 0.009. (a)If the respondents answer truthfully, what is P(0.49 ≤ V ≤ 0.53)? This is the probability that the sample proportion V estimates the population proportion 0.51 within plus or minus 0.02. (b)In fact, 49% of the respondents said they planned to vote for Barack Obama (V = 0.49. If respondents answer truthfully, what is P(V ≤ 0.49) ?

12.55 Playing Pick 4. The Pick 4 games in many state lotteries announce a four-digit winning number each day. Each of the 10,000 possible numbers 0000 to 9999 has the same chance of winning. You win if your choice matches the winning digits. Suppose your chosen number is 5974. (a)What is the probability that the winning number matches your number exactly? (b)What is the probability that the winning number has the same digits as your number in any order?