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AP Statistics Chapter 5 Probability: What are the Chances? 6.1 – The Idea of Probability • Chance behavior is unpredictable in the short run, but it has a regular and predictable pattern in the long run. 2 Randomness • We consider an event “random” if individual outcomes are uncertain, but there is a regular distribution of outcomes in a large number of repetitions. • The actual probability of any outcome of any random phenomenon is the proportion of times that the outcome occurs in a long series of repetitions. • This can also be considered the “long-term relative frequency”. 3 Simulating Experiments • Sometimes it is much more efficient to simulate an experiment than actually going through the entire process. • A simulation is the imitation of chance behavior based on a model that accurately reflects the experiment under consideration. • The key to simulation is to make use of a trustworthy model. • Models can be developed through use of random numbers, computers, calculators, and even dice or spinners. 4 Steps to Simulating • • • • • State the problems or describe the experiment. State the assumptions. Assign digits to represent outcomes. Simulate many repetitions. State your conclusions. 5 EXAMPLES • Use the random number table (start with line 110) to simulate the number of heads in 10 flips. Get twenty counts! • Use the calculator to complete the same simulation. • Kobe Bryant has made 83% of the free throws he has shot during his NBA career. Simulate the number of makes that he would have in shooting 10 free throws, 15 times. • Do this with a number table on line 112. • Repeat this experiment with the calculator. 6 One More Example • A cereal company runs a promotion in which each box of cereal will contain a collectible card with one of these NASCAR drivers: Jeff Gordon, Dale Earnhardt Jr., Tony Stewart, Jimmie Johnson, or Danica Patrick. • The company claims that each of the 5 cards is equally likely to appear in a box of cereal. • A NASCAR fan is very surprised when it ends up taking her 23 boxes to get the full set of cards. • Does she have a reason to dispute the company’s claim of equal distribution? 7 Now, Simulate… • STATE: What is the probability that it will take 23 or more boxes to get all 5 cards? • PLAN: Come up with 5 numbers to represent the cards. Let 1= Jeff Gordon, 2 = Dale Earnhardt Jr., 3 = Tony Stewart, 4 = Jimmie Johnson, and 5 = Danica Patrick. Now use the RandInt function on the calculator to simulate buying a box of cereal. • DO: Now run this until you “collect” all five cards. Repeat this process. • CONCLUDE: How common would it be to take 23 tries to get all cards? 8 HOMEWORK Complete the problems: pg. 293 (#1 – 20). This assignment will be due for completion at the start of the next session of class. Probability Models • All probability models must include: • A list of all possible outcomes • A probability for each possible outcome • The set of all possible outcomes of a random phenomenon is referred to as the SAMPLE SPACE. • Any individual or set of outcomes is called an EVENT. 10 Sample Space • If we were to consider the sample space of rolling two dice, what would it be? 11 EXAMPLE • Using the previous information, what would be the theoretical probability of rolling a 5? • Compute an entire probability model for the sum of the two dice. 12 Multiple Events Combined • What would be the sample space for the combined events of flipping a coin and then rolling a single die? • In such a case, we may find it easier by making a tree diagram. • The tree diagram will show what each possible event in the sample space is. 13 TREE DIAGRAM • The tree diagram for flipping a coin and rolling a die looks like this: 14 Multiplication Principle • Another method of computing the number of outcomes in a sample space is to use the multiplication principle. MULTIPLICATION PRINCIPLE If there are “a” different ways to do a task and there are “b” different ways to do a 2nd task, then there are a X b ways to combine the events. If a menu offers a choice between 3 meats and 5 vegetables, how many different meals could be ordered? 15 Replacement • It is also very important to note if a situation will allow replacement. • Replacement means placing the selected item back in to the total before selecting again. EXAMPLE How many three digit numbers can we make? 10 x 10 x 10 How many can we make without replacement of digits? 10 x 9 x 8 16 Probability Rules 1. All probabilities are between 0 and 1. 0 P( A) 1 2. All outcomes will have a sum of probabilities equal to 1. P( S ) 1 3. The probability that an event DOES NOT occur is 1-minus the probability that it does. 4. If two events haveP no(outcomes A ) 1in common, P( Athe ) probability that one OR the other occurs is the sum of their individual probabilities. c P( AorB) P( A) P( B) 17 UNION? • The union of two events is sometimes symbolized as: A B • This is read as “A union B”, and it represents the set of all outcomes in either A or B. A B ( A or B) 18 INTERSECTION • The intersection of two events is symbolized as: A of B • This is read as the intersection A and B, and it represents all outcomes that are both A and B. • If (A and B) is empty, then • This would mean that A and B are DISJOINT or MUTUALLY EXCLUSIVE. A B • This allows us to use the formula: P( AorB) P( A) P( B) 19 Check Your Understanding Complete the Check Your Understanding problem on the top of pg. 303. We will discuss the answers in a moment. 20 HOMEWORK Complete the problems: pg. 309 (#39 – 50; 52 - 54). This assignment will be due for completion at the start of the next session of class. Conditional Probability • Conditional Probability is denoted by P(A|B) • This would be read as the “Probability of A, given that B has occurred. • If we are playing poker (5 card-stud) and have four cards dealt, one of which is an ACE… • What is the probability of getting another ACE on the fifth card? • P(ace | 1 of 4 cards is an ace) = 3 1 48 16 22 Conditional Probability • When P(A) > O, then we can use the formula: P( A and B) P( B | A) P( A) • What is the conditional probability that a woman is a widow, given that she is at least 65 years old? 23 Independence • We can use conditional probability to define independent events. • Two events are independent if… P( B | A) P( B) • In other words, the probability of B will be the same whether A occurs or not. 24 HOMEWORK Complete the problems: pg. 329 (#63 – 76). This assignment will be due for completion at the start of the next session of class. General Multiplication Rule • The general multiplication rule applies to find the probability of the intersection of two events. • The probability that events A and B both occur can be found by: P( A and B ) P( A B ) P( A) P( B | A) • Remember that P(B|A) = P(B) if A and B are independent. 26 EXAMPLE The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the internet, and that 55% of online teens have posted a profile on a socialnetworking site. What percent of teens are online and have posted an a profile? 27 EXAMPLE Cont. Here is a tree diagram that models the relationship. P(online and have profile) P(online) P(profile|online) (0.93)(0.55) 0.5115 In other words, about 51% of all teens use the internet AND have a profile on a social site. 28 Another Example Video-sharing sites, like YouTube, are popular destinations on the internet. Looking only at adult internet users, about 27% of all adult users are 18 to 29 years old. Another 45% of the users are 30 to 49 years old. The remaining 28% are 50 or older. It has been found that 70% of the users aged 18-29 regularly visit video-sharing sites, along with 51% of those aged 30-49, and 26% of those 50 or older. Do most users visit these sites? 29 EXAMPLE Cont. We should start by restating all of the probabilities that we have been given. P(age18 to 29) 0.27 P(age 30 to 49) 0.45 P(age 50+) 0.28 P(vid yes| 18 to 29) 0.70 P(vid yes| 30 to 49) 0.51 P(vid yes| 50+) 0.26 We are trying to find P(vid yes), as it applies to the entire population. Let’s look at a tree diagram… 30 Example Cont. P(18 29and vid yes) (0.27)(0.70) 0.1890 P(30 49and vid yes) (0.45)(0.51) 0.2295 P(50 and vid yes) (0.28)(0.26) 0.0728 P(vid yes) 0.1890 0.2295 0.0728 0.4913 31 Check Your Understanding Complete the Check Your Understanding problem on the top of pg. 321. We will discuss the answers in a moment. 32 HOMEWORK Complete the problems: pg. 330 (#77 - 89). This assignment will be due for completion at the start of the next session of class.