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June 16 Day 6 The Addition Read pp. 274 - 281 due 5.2 Rule and Do pp. 281 - 285, # 1 – 47 (odd) June 21 Complements Vocabulary: { disjoint, mutually exclusive, contingency table, row variable, column variable, cell, complement of an event } Independence and the Read pp. 286 - 289 due 5.3 Multiplication Do pp. 290 - 292, # 1 - 31 June 21 Rule Vocabulary: { disjoint events, independent events, At-least probabilities } Addition Rule for Disjoint Events Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. What does “S” stand for? 1 Example: Using the Venn Diagram on Page 1: (A) P(Red) = (B) P(0) = Addition Rule for Disjoint Events If E and F are disjoint (or mutually exclusive) events, then P E or F P E P F note: The Addition Rule also works with more than two mutually exclusive events. Example: Benford‟s Law When we write numbers, we do not write them with a “0” in the first digit, i.e., 25 is not 025. Thus, numbers begin with 1 through 9. We might expect that the first digit of the numbers we use would begin with equal frequency, all numbers appearing 1/9 of the time. Physicist Frank Benford was credited with stating that they do not. In fact, he asserted that they occur as shown in the table below: 2 (A) Verify that Benford‟s Law is a probability model (B) Use Benford‟s Law to determine the probability that a randomly selected first digit is 3 or 6 (C) Use Benford‟s Law to determine the probability that a randomly selected number is greater than 5. Example # 2a: Using a Standard Deck of Cards (52 and no Jokers) (A) Compute the probability of the event E = “drawing a 9” (B) Compute the probability of the event E = “drawing a 9” or “drawing a 3” (C) Compute the probability of the event E = “drawing a 9” or “drawing a 3” or drawing a Queen” Example #2b: If A and B are mutually exclusive events with P(A) = .33 and P(B) = .25, what is the P(A or B)? 3 The General Addition Rule For any two events E and F: P E or F P E P F P E and F Example #3a: A two-question survey was taken of Tri-C. “Do you like chocolate ice cream?” and “Do you like vanilla ice cream?” E = Chocolate F = Vanilla P E 42% P F 20% P E and F 14% Find P E or F 4 Example #3b: The registrar‟s office searched student records to find what percent of the freshman were enrolled in a math class. A second search discovered what percent of the freshman were taking an English class. M = math class E = English class P M .66 P M or E .92 P M and E .48 Find P E Example #3c: The guidance department totaled the results from Freshman surveys. The two largest areas of career interests were medical and business. M = medical B = business P M .23 P B .25 P M or B .34 Find P M and B 5 Contingency Tables (or two-way tables) Example #4: A furniture store did a survey of 180 customers regarding customer satisfaction as well as how the customer learned of the store. TV ad Walk-In Referred Total Not Satisfied 5 10 4 19 Neutral 12 9 19 40 Satisfied 22 18 28 68 Very Satisfied 15 14 24 53 Total 54 51 75 180 Assume the sample represents the entire population of customers. Find the probability that a customer is: a) Learned of the store through a TV ad. b) Very satisfied c) Arrived at the store as a Walk-In or Referred d) Was Satisfied or Referred e) Satisfied and Referred 6 The sum of the probabilities of all simple events in a sample space must equal 1. The complement of event A is the vent that A does not occur. Ac designates the complement of event A. Furthermore, 1. P E P Ec 1 2. P event E does not occur P E c 1 P E Example #6a: Tri-C has renovated one of the classrooms and installed „captains‟ chairs that are both cushioned and swivel. The interior designer also thought it would be appealing to install chairs with different colors. In one classroom, the following colored chairs were installed: Green Blue Red Yellow Black 8 4 6 7 5 The professor announces that he will select a color and then all students who are seated in those colored chairs will have to make a presentation from the homework. You are seated on a green chair and are worried because last night you went to the Indians game and didn‟t do your homework. What is the probability that you will have to make a presentation? What is the probability that you won‟t have to make a presentation? Example #6b: Playing roulette. The primary pockets on a Roulette wheel are numbered from 1 to 36 with each alternating between red and black. There also is a green 0 and in the American Roulette another green 00. Using the Law of Large numbers, if you played the game 2000 times selecting only RED on each play, how many times would you expect to win? 7 Independence and the Read pp. 286 - 289 due 5.3 Multiplication Do pp. 290 - 292, # 1 - 31 June 21 Rule Vocabulary: { disjoint events, independent events, At-least probabilities } Two events are independent if the occurrence or nonoccurrence of one does not change the probability that the other will occur. Two event are dependent if the occurrence or nonoccurrence of one does affect the probability that the other will occur. Example # 1 Independent or Not? A) A fair coin is tossed four times and it comes up HEADS all four times. Your friend says that the next toss will most likely come up TAILS. Right or Wrong? B) A survey of people residing in Cuyahoga County in 2009 has several categories. Two of the categories are: “Attends Private College/University” and “Family Income greater than $100,000”. Are these independent? C) Twin sisters Eva and Deeva are 21. One auditions and is selected for the Cavaliers Dance Team and the other auditions and is selected for The Ohio State Buckeyes Cheer Team. Are these independent? Disjoint Events are not the same as Independent Events These may seem at first glance to be the same. However, two events are disjoint if they have no outcomes in common. Independent events means that one does not affect the other. E = “Born in a month ending in Y” F = “Born in a month not ending in Y” E and F are disjoint. J = “Mrs. Brown has a baby in July” K = “Mrs. Brown‟s Grandmother goes on a cruise.” 8 Multiplication rule for independent events: P E and F P E P F Example #2a: A fair coin is tossed at the same time that a die is rolled. What is the probability that the coin will land heads up and the die will show a 4? Example #2b: A fair coin is tossed at the same time that a die is rolled. What is the probability that a coin will land tails up and the die will show a number greater than 4? The Multiplication rule can be extended to more than two events: Multiplication rule for n independent events: P E and F and G and ... P E P F P G ... Example # 3: The Nevada Gambling Commission requires that odds for slot machines must be posted on the actual machine. Usually they are posted on the back which is pressed against a wall of the back of another slot machine. (It is strongly recommended that you do not attempt to pull the slot machine out to verify this!) The odds vary. We are about to play a slot machine in Casino del Luthy. The odds are 47%. a) What are the odds that you will win on the first try? b) What are the odds that you will win on both the second and third try? c) What are the odds that you would win on three consecutive tries? 9 Example 4: Black Jack or “21” is one of the very few games of chance where a smart gambler can tip the odds in his favor. In order to do this, the gambler must be aware of which cards have already been played and what are the probabilities for the cards still to be dealt. In a deck of 52 cards, there are four Aces. What is the probability that both of the first two cards dealt will be twos? Example 5: The tour director at the Smelly Jelly Bean Company was a former statistics teacher. He like to fill a ceramic jar (non-transparent) with candies and then let guests at the factory reach in and take out a candy. He then asked them which color had the largest number of beans. Below is a table of how he recently filled the jar. Color Blue Green Red Yellow Brown Black Number 31 24 40 17 50 33 A guest will get to select one bean. The statistics teacher will replace whatever color is chosen with the same color before the next guest has an opportunity. a) What is P(red)? b) What is P(green) followed by P(yellow)? c) What is the P(brown) followed by P(black) and another P(brown) The tour director has run out of replacements and no longer can put additional candies in the jar. d) What is P(blue) followed by P(green)? e) What is P(black) followed by P(black)? 10 Computing At-Least Probabilities: The concept at play with “At-Least” is the complement rule. The complement of event A is the vent that A does not occur. Ac designates the complement of event A. Furthermore, 1. P E P Ec 1 2. P event E does not occur P E c 1 P E Example # 4: Mrs. Tell, expert bow-and-arrow marksman, hits her target 95% of the time from a distance of 50 feet. Suppose one day she is baby-sitting the six Brady-Bunch children and decides to play “Shoot the Apple Off Your Head”. The eager children willingly line up 50 feet away with apples on their head. What is the probability that at least 1 time she will miss the apple? Method 1: P(1 or 2 or 3 or 4 or 5 or 6) = P (1 miss) + P (2 misses) + P (3 misses) + … This is time consuming and difficult. In fact, we really don‟t have the mathematical background at this time to make the calculations. We will learn about it in Lesson 6.2. Method 2: P (at least one miss) = 11

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posted: | 8/30/2011 |

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