Chapter 6: Counting, Probability and Inference

Transcription

1 Chapter 6: Counting, Probability and Inference 6.1 Introduction to Probability Definitions Experiment a situation with several possible results o Ex: Outcome each result of an experiment o Ex: Sample Space the set of all possible outcomes of an experiment o Ex: Event any subset of the sample space of an experiment o Ex: Ex 1: Two six-sided dice, one red and one green, are thrown, and both numbers are recorded. a. List all possible outcomes for the experiment. b. How many outcomes are in the sample space for this experiment? Ex 2: A small box contains 30 blue, 30 green and 25 red paper clips. Two paper clips are taken from the box and their colors recorded. a. List all possible outcomes for this experiment. b. How many outcomes are possible in the sample space for this experiment? Probability of an event Always a number between and Measures the chance that an will occur Can be written as: o o o Definition of Probability

2 Ex 3: An experiment consists of tossing two fair coins and counting the number of heads. a. Find the probability that you will toss 1 head. b. Find the probability that you will toss 2 heads. c. Find the probability that you will toss 0 heads. d. Are these events equally likely? Ex 4: A researcher is studying the number of boys and girls in families with three children. Assume that the birth of a boy or a girl is equally likely. a. List the sample space b. Find the probability that a family of three children has exactly one boy c. Find the probability that a family of three has at least 2 girls d. Find the probability that a family of three has all boys Ex 5: You toss two fair die. What is the probability that the sum is greater than 8? Empty Set or Null Set Basic Properties of Probability Let S be the sample space associated with an experiment. Then, for any outcome or event E in S,

3 6.2 Principles of Probability Mutually Exclusive/Disjoint VS Inclusive Union of Events Activity 1: a. Toss a pair of fair six-sided dice twenty times, recording the sum of face-up numbers each time. b. Use the frequency table to record each sum using the combined data for your class. c. a. What was the class s relative frequency of having a sum of 7? b. What was the class s relative frequency of having a sum of 11? c. What was the class s relative frequency of having a sum of 7 or 11? Sum Frequency Theoretical Probability d. Which of the events in Step 3 are mutually exclusive? Activity 2: Fill in the column for theoretical probabilities for all of the sums. Use those probabilities to answer the following questions. 1. Which sums are prime? 2. Find the probability that the sum is prime or odd? Are these events mutually exclusive or inclusive? 3. Find the probability that the sum is even or divisible by 3. Are these events mutually exclusive or inclusive? 4. Find the probability that the sum is 4 or 6. Are these events mutually exclusive or inclusive?

4 The following table summarizes that language of the set of intersections Set Language Disjoint Intersecting Event Language Venn Diagram Symbols Ex 1: Suppose at a high school, 298 students study only French, only Spanish, or both languages. The school reports 115 students study French and 209 study Spanish but because > 298, there must be students who study both languages. How many students study both? Ex 2: A pair of six-sided dice is thrown. If the dice are fair, what is the probability that the dice show doubles or a sum less than 10? Ex 3: A pair of six-sided dice is thrown. If the dice are fair, what is the probability that the dice show a sum of 7 or 11? Ex 4: A pair of fair six sided dice is thrown. What is the probability that exactly one die shows a 3 or the sum of the numbers is greater than 9? Complimentary Events When events are mutually exclusive and their union is the sample space The is called Ex: Experiment Sample Space Event Complement Tossing a coin Tossing two coins Probability of Complements

5 Ex 5: Refer to the students studying languages in Ex 1. If a student is selected at random, what is the probability that he is not studying both languages at the same time? Ex 6: A pair of six-sided dice is thrown. If the dice is fair, what is the probability that the dice show a product between 7 and 31? Independent Events If the probability of A does affect the probability of B Ex 7: A coin is tossed and a six-sided die is rolled. Find the probability of getting a head on the coin and a 6 on the die. Ex 8: A jar of marbles contains 4 blue marbles, 5 red marbles, 1 green marble and 2 black marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. Find the probability of the following. a. P(green and red) b. P(blue and black) Ex 9: A jar of marbles contains 4 blue marbles, 5 red marbles, 1 green marble and 2 black marbles. A marble is chosen at random from the jar. You keep that marble out of the jar, then choose a second marble. Find the probability of the following. a. P(green and red) b. P(blue and black)

6 6.3/6.4 Counting Strings with/without Replacement The Counting Principle Ex 1: A popular game show features a spinner divided into twenty-four congruent sectors and numbered something like the wheel shown at the right. The spinner cannot stop on a boundary line. You spin it twice. Determine the number of elements in the sample space S. Ex 2: A lottery game played in some states involves picking 3 digits from 0 to 9. Determine the number of elements in the sample space. Ex 3: a. On a 28 questions multiple-choice mathematics test, each question has 5 choices. How many possible completed answer sheets are there? b. If you guess randomly on each question, what is the probability of answering all 28 questions correctly? Ex 4: In a certain state, license plates have two letters followed by 4 digits from 0 through 9. How many license plates are possible? Permutations Ex 5: Simplify 9P 5

7 Ex 6: The Big 10 athletic conference consists of 11 teams: Illinois, Indiana, Iowa, Michigan, Michigan State, Minnesota, Northwestern, Ohio State, Penn State, Purdue, and Wisconsin. You want to predict which team will finish first in a particular sport, which second, and which third. How many different predictions are possible? Ex 7: In volleyball, a team plays 6 players at a time, three at the net and three back row. How many different ways are there to arrange three at the net from the six who are playing? Ex 8: There is an elementary school classroom that has 6 girls and 11 boys in it. a. How many different ways can all 17 students line up in a line? b. How many ways can the students line up if the girls have to go before the boys? c. In how many ways can the student s line up if the line must be boy, girl, boy, girl etc. until there are no more girls to go in between the boys? d. How many ways can the student s line up the tallest person has to be in the front of the line and the shortest person has to be in the back of the line? Ordering Duplicate Objects Ex 9: How many ways are there to order the letters a,a,b? Ex 10: How many ways can you order the letters in the word basketball?

8 6.5 Contingency Tables Tables that divide outcomes among two or more variables are called Ex 1: Refer to the Titanic table below. Round to the nearest tenth of a percent. a. Out of all people on the Titanic, what percent survived? b. Find out the percent of passengers in first class who survived. c. Find the percent of passengers who survived that we in first class. Tree diagrams can help to clarify the difference between parts A and B Other Contingency Table Formats Ex 2: A 2001 study by the University of Texas Southwestern Medical Center examined 626 patients to see if there was a connection between getting a tattoo and infection with Hepatitis C (HCV). The results are in the contingency table below. a. Add row and column totals to the table. b. What does the total of the third column represent? c. What does the total of the second row represent? d. According to this data, is someone with a tattoo done in a commercial parlor more or less likely to have HCV than someone with a tattoo done elsewhere? e. Give at least one reason why the result in part d might not reflect the safety of each kind of tattoo?

9 Column/Row Percents Take a look at the following variations on the table we saw in example 1 Ex 3: Fifth-grade students in a school were surveyed about their favorite book series. The results are reported in the contingency table below. a. What does the 23% in the first row, third column represent? b. Which is larger: the number of boys who prefer Animorfs, or the number of girls who prefer Lemony Snicket? 6.6 Conditional Probability Refer to the titanic table below. Let A and B be events with A = the passenger was in second class and B = the passenger survived. We want the probability of. Such probability is called. Definition of Conditional Probability The conditional probability of an event B given an event A, written

10 Ex 1: An article in the Journal of the American Medical Association in 1997 reported that, when people go to their doctor s office with a sore throat and think they might have strep throat, 30% actually have strep throat. It noted that a current test for strep throat was 80% accurate if you have strep throat and 90% accurate if you do not. What is the probability that a person who receives a positive result from this test does not have the disease? Ex 2: Let B = a person eats a good breakfast; let L = a person eats a good lunch. Suppose, in a group of 80 people, 43 eat good breakfasts and good lunches, 21 eat good breakfasts but not a good lunch, 12 eat a good lunch but not a good breakfast, and the rest eat neither a good lunch nor a good breakfast. a. Find P(B L) b. Find P(L B) c. Find P(B L) Ex 3: Suppose all patients are tested for a serious disease that is estimated to be found in 0.5% of people. Suppose also that the test accurately spots the disease 98% of the time and accurately indicates no disease 95% of the time. What is the probability that a negative result is a false negative? Ex 4: Suppose that 1 in 500 airline passengers carry some hazardous material on them when on a plane. Further suppose that an airport screening devise accurately identifies 98% of people with hazardous materials that pass through it, and accurately identifies 99% of people without hazardous materials. If a person is identified by the machine as having hazardous materials, what is the probability that the person actually has these kinds of materials?