8.3 Probability with Permutations and Combinations

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1 8.3 Probability with Permutations and Combinations Question 1: How do you find the likelihood of a certain type of license plate? Question 2: How do you find the likelihood of a particular committee? Question 3: How do you find the probability of winning a lottery? Question 4: How do you find the likelihood of detecting a defective product? In sections 8.1 and 8.2, you learned about counting objects using the Multiplication Principle, permutations, and combinations. With these strategies, we are able to count the number of different license plate numbers or ways to select lottery numbers. In this section you ll go one step further and use these strategies to find probabilities. The key to finding these probabilities is an assumption. We will assume that the outcomes in whatever event and experiment we are considering are equally likely. This will enable us to calculate the probability of an event by counting the number of outcomes in it. Specifically, we ll find the probability of an event E from a sample space S with P E n E n S where ne and ns are the number of outcomes in E and S. This will helps us to calculate the likelihood (or unlikelihood) of events such as winning the lottery jackpot or detecting a defective product on a production line. 1

2 Question 1: How do you find the likelihood of a certain type of license plate? Problems involving personal identification numbers, passwords, and license plates usually require the use of the Multiplication Principle. You might be tempted to use permutations, but letters and numbers may generally be repeated in these types of problems. In fact, two of the most common ATM PINs are 1111 and In permutations, the repetition of numbers is not allowed. Example 1 License Plate Numbers Many states offer license plates which a person may choose the letters and numbers on it. Typically, people choose words or phrases with some meaning for them. Sometimes randomly generated license plates have sequences of numbers or letters that might be offensive or have a separate meaning. If the license plates in a particular state consist of three letters followed by three numbers, what is the probability that a randomly generated plate ends in the numbers 911. Solution Suppose the experiments is choosing a license plate randomly that consists of three letters followed by three numbers. In Section 8.1, we found the total number of possible license plate with the Multiplication Principle, first second third first second third letter letter letter number number number 17,576,000 To find the probability that a license plate ends with 911, we also need to know how many license plates end with 911. Applying the Multiplication Principle again gives first second third first second third letter letter letter number number number 17,576 2

3 There is only 1 way to make the last three choices since the numbers must be 911. The probability of a license plate ending with 911 is P plate ends with "911" n plates ending with "911" n plates with proper format 17,576 17,576, In problems like this example, the Multiplication Principle is used since letters and numbers on the plate may be repeated. 3

4 Question 2: How do you find the likelihood of a particular committee? In Section 8.2, we counted the number of ways to create an executive leadership team. Since this was simply a grouping of executives and each team member was different, combinations were used to count the number of ways to put together the team. Example 2 Team Selection A large nonprofit corporation wishes to form an executive leadership team from a group of 6 male executives and 8 female executives. The team will have 6 members. If the team members are selected randomly, find the probability of each team described below. a. The team has equal numbers of men and women. Solution The experiment for this example is the random selection of a team of six. A team with equal numbers of men and women will have three men and three women. Since rearranging team members leads to an identical team, combinations are used to count the teams. The number of six person teams selected from the fourteen executives is 14! C 14, !6! teams The number of teams with equal numbers of men and women is calculated using the Multiplication Principle. The choices are choosing the men and choosing the women. This gives the number of teams, Choose 3 members from 6 men C Choose 3 members from 8 women C 6,3 8, teams Choose Choose 3 men 3 women The probability of randomly selecting a team with equal numbers of men and women is 4

5 P Team with 3 men and 3 women n Teams with 3 men and 3 women n b. The team has more than four women. Teams with 6 members Solution Teams with more than four women must have five women and one man or six women and no men. The number of teams with five women and one is C C 8,5 6, teams Choose Choose 5 women 1 man The number of teams with six women and no men is C C 8, 6 6, teams Choose Choose 6 women no men To count the number of team in the event team has more than 4 women, add the number of teams in these two team compositions, Teams with 5 women and 1 man Teams with 6 women and no men n Teams with more than 4 women teams The probability of randomly selecting a team with more than four women is P Team has more than 4 women n Teams with more than 4 women nteams with 6 members

6 Question 3: How do you find the probability of winning a lottery? Many people buy their weekly lottery ticket with the hope of getting rich quick. A dollar or two a week does not seem like a high price to pay for a chance at winning millions. The Powerball lottery played in 42 states recently had a jackpot of almost $600 million dollars that was split by two winners. In the Mega Millions lottery game, you pick five numbers from the 1 through 56 and another number (from the Mega Ball) from 1 through 46. If all of the numbers match the numbers drawn by lottery officials, the player wins the jackpot. To find out the number of different equally likely outcomes there are to randomly drawing the numbers. think of the selection of the numbers as two choices. First, select the first five numbers from the 56 numbers. Since the order in which the numbers are drawn do not make a difference, this can be done in 56! C 56,5 3,819, !5! ways Second, choose the last number from the 46 numbers. This can be done in 46 ways. The total number of different ways the numbers may be picked is the product of the numbers of ways each choice may be done, C(56,5) 46 3,819, , 711,536 ways Choose five numbers Choose last number Each of these ways is equally likely. If we consider the experiment to be picking lottery numbers and the event A to be matching all of the numbers, the probability of matching all six numbers is P A n A n S 175,711,536 In the example below, we examine another set of numbers that results in a much lower payoff in the Mega Millions Lottery. 6

7 Example 3 Lottery In the Mega Million Lottery, a player wins $250,000,000 if they match five numbers, but not the Mega Ball number. Find the probability of winning $250,000 in the lottery. Solution We have already calculated the number of ways to select the numbers for the Mega Millions Lottery, 175,711,536 ways. To find the number of ways to match five numbers and not the Mega Ball number, break the selection of balls into two choices. First, choose the numbers that match the numbers selected by lottery officials. Since there is only one set of numbers that match, there is only one way to match the first five numbers. Second, choose the Mega Ball number os that is does not match. There are 46 Mega Ball numbers so 45 of the numbers will not match. Apply the Multiplication Principle to these choices to give Match first Do not match 5 numbers Mega Ball The probability of matching five numbers, but not the Mega Ball is P Match 5 numbers 175,711, n Match 5 numbers n Select 6 numbers 45 Compared to the likelihood of winning the jackpot, it is 45 times more likely to match the first five numbers. However, this liklelihood is still very small. As a player, this is more of an unlikelihood than a likelihood! 7

8 Question 4: How do you find the likelihood of detecting a defective product? When a product is manufactured, it is possible that the production process may lead to some products being defective. A low number of defective products is acceptable, however a high number leads to a high number of warranty claims. When the manufacturer packages the products, some of the defective products may be in the package. We can calculate the likelihood that a package will contain no defective products using combinations. The goal is to make this probability as low as possible in order to minimize the warranty costs. Example 4 Quality Control As the number of electronic devices increases, so does the use of rechargeable batteries. A particular manufacturer produces batteries in lots of 100. In each lot, two of the batteries will be defective. The batteries are randomly packaged in groups of four batteries. What is the probability that all of the batteries in a package will not be defective? Solution The order in which the batteries appear in a package is irrelevant, so combinations are used to calculate the number of ways to package the batteries. The number of ways to select four batteries from the lot of 100 is C n 4 batteries from lot 100,4 3,921,225 A package with no defective batteries will have batteries selected from the 98 batteries in a lot that are not defective. The number of ways to select 4 nondefective batteries from the 98 nondefective batteries is C n 4 nondefective batteries 98, 4 3, 612, 280 The probability of a package having four batteries that are not defective is 8

9 P 4 nondefective batteries n 4 nondefective batteries n 4 batteries from lot 3,612,280 3,912, If we know the likelihood of having no defective batteries in a package of four, we can also calculate the probability of the compliment of this event. The compliment of four batteries that are not defective is at least one battery is defective. This probability is P At least one battery defective1p4 nondefective batteries If the cost to warranty batteries is high, this probability may be too high. Companies lower this probability by improving their manufacturing processes. This reduces the number of defective units in each lot. 9

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