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Part 4: Counting

Part 4: Counting. High. http://brownsharpie.courtneygibbons.org/?cat=22. BCR – Example 1. In this class there are Sophomores (S), juniors (J) or Seniors ( Sr ). In addition, each person can be either male (M) or female (F). How many different possibilities are there ?

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Part 4: Counting

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  1. Part 4: Counting High http://brownsharpie.courtneygibbons.org/?cat=22

  2. BCR – Example 1 In this class there are Sophomores (S), juniors (J) or Seniors (Sr). In addition, each person can be either male (M) or female (F). • How many different possibilities are there? • How many different possibilities would there be if there were Math Ed majors, Biology majors and Statistics minors also in the class?

  3. BCR – Example 2 This semester, there are 16 students in STAT 311 with 3 females and 13 males. If I choose 3 students at random with replacement, what is the probability that the first student is female, the second student is male and the third student is male?

  4. Ways of Counting

  5. Sampling with Replacement, order matters (BCR) Suppose that a sample of size 2 is drawn with replacement from a population of size 5. a) Use a direct listing to determine the number of possible ordered pairs. b) Solve part (a) by using BCR. c) Determine the number of possible ordered samples of size r with replacement from a population of size n.

  6. Sampling Without Replacement, order matters (Permutation) Suppose that a sample of size 2 is drawn without replacement from a population of size 5. a) Use a direct listing to determine the number of possible ordered pairs. b) Solve part (a) by using BCR.

  7. Permutations - Example • How many possible ways can you draw an ordered sample of size 0 from a population of size n? • How many possible ways can you draw an ordered sample of size n from a population of size n?

  8. Sampling Without Replacement , order does not matter(Combination) Suppose that a sample of size 2 is drawn without replacement from a population of size 5. Use a direct listing to determine the number of possible unordered pairs.

  9. Combinations - Example • How many possible ways can you draw an unordered sample of size 0 from a population of size n? • How many possible ways can you draw an unordered sample of size 1 from a population of size n?

  10. Sampling With Replacement, order does not matter (SB) Suppose that a sample of size 2 is drawn with replacement from a population of size 5. a) Use a direct listing to determine the number of possible unordered pairs. b) Determine the number of possible unordered samples of size r with replacement from a population of size n.

  11. BCR: Examples • Consider the random experiment of rolling one 4-sided die. If the die is rolled 2 times, how many possible outcomes are there? • Arizona plates consist of three digits followed by three letters. • How many different license plates are possible? • What is the probability that a particular license plate doesn’t have any repeating digits or letters?

  12. Permutation: Examples • How many ways can I select the order of the top 3 students in this class (class size = 16)? • How many different ways can I choose 5 dice from a bag of 20 dice without replacement? • How many ways can we arrange 9 dice? • What happens if out of the 9 dice, we have 3 white dice, 4 speckled red dice and 2 plane red dice, if we want to keep the different types of dice together, how many possible arrangements are there?

  13. Combination: Examples • In an attempt to attract people to buying Kindle books, a merchandiser states that if a person buys 4 Kindle books, that person will get two free. Currently, this merchandiser has 60 Kindle books in stock. How many possibilities does the person have to selecting the 6 books? • The IRS decides that it will audit the returns of 3 people from a group of 18. If 8 of the people are women, what is the probability that all 3 of people audited are women?

  14. SB: Examples • How many different sets of non-negative numbers x, y and z are solutions for the following equation: x + y + z = 136. • How many ways are there to buy 13 bagels from 17 types if you can repeat the types of bagels?

  15. Example An ordinary deck of 52 playing cards is shuffled and dealt. What is the probability that a) the 7th card is an ace? b) the 7th card is the first ace?

  16. Ordered Partition - Definition An ordered partition of n objects into r distinct groups of sizes n1, n2, …, nris any division of the n objects into a combination (unordered) of n1objects in the first group, n2objects in the second group, etc. This number is denoted by

  17. Ordered Partition: Example • List all of the possible ordered partitions of these 5 letters into two distinct groups of sizes 3 and 2. b) Use part (a) to determine the number of possible ordered partitions of the 5 letters into the two groups. c) Use the combinations rule and BCR to determine the number of possible ordered partitions of the 5 letters into the 2 groups.

  18. Ordered Partition: Example 2 (class) • If you want to partition your class of 30 students into 7 groups, how many possible ways can you do this if the group sizes are 3, 4, 5, 5, 5, 4, and 4? • How many different possible hands are there in a 5-card draw poker game with 5 players?

  19. Birthday Problem • What is the probability that at least two students in this class, size = 21, have the same birthday? • What is the probability that at least two students in this class, size = 30, have the same birthday?

  20. Coincidences …Once we set aside coincidences having apparent causes, four principles account for large numbers of remaining coincidences: hidden cause; psychology, including memory and perception; multiplicity of endpoints, including the counting of "close" or nearly alike events as if they were identical; and the law of truly large numbers, which says that when enormous numbers of events and people and their interactions cumulate over time, almost any outrageous event is bound to occur. These sources account for much of the force of synchronicity. (Abstract) ….. The probability problems discussed in Section 7 make the point that in many problems our intuitive grasp of the odds is far off. We are often surprised by things that turn out to be fairly likely occurrences. (Introduction) Diaconis, P. and Mosteller, F. "Methods for Studying Coincidences." J. Amer. Statist. Assoc.84, 853-861, 1989

  21. What method to use? • How many batting orders are there for 9 players on a baseball team? [362,880] • How many ways can a person draw three dice from a bag containing identical 10 dice if after each draw the person puts the drawn dice back into the bag. [220] • In a math club at Purdue with 20 members, 3 people can go to a national conference. How many different ways can these people be chosen? [1140] • Out of 10 people in the U.S., 2 live in the Northeast, 3 live in the Midwest, 3 live in the South, and 2 live in the West. How many different ways can these 10 people be assigned to these regions? [25,200]

  22. What method to use? (cont) • At a movie festival, a team of judges is to pick the first, second, and third place winners from the 18 files entered. How many possible ways are there to choose the winners? [4896] • If a password contains 7 lower case letters, how many possible passwords are there? [8,031,810,176] • Suppose that a small pond contains 500 fish, 50 of them tagged. A fisherman catches 10 fish which he cannot tell apart. After each time he catches a fish, he throws the fish back into the pond. Find the number of ways that he can catch 2 tagged fish. [1275] • In 5-card draw, how many different hands can you receive? In 5-card draw, the order of the cards is not important. [2,598,960]

  23. What method to use? (cont) • For a certain airport, there are three runways that are used, A, B and C. Out of the next 15 jets, how many ways can 10 land on runway A, 2 land on runway B and the rest land on runway C? [30,030] • A multiple choice exam has 40 questions each of which have 5 possible choices. How many possible combinations of answers are there? [9.09 X 1027] • You are groups of 5 students each from your class of 20 students. If each of these groups are performing different assignments, how many ways can you arrange your class? [1.17 X 1010] • A statistics professor wants to do perform in depth interviews to see how she is teaching. So she has decided to choose 5 students from her class of 40. How many different possibilities are there? [658,008]

  24. What method to use? (cont) • The menu at a restaurant has five choices of a beverage, three different salads, six entrées and four deserts. How many different meals are possible? [360] • How many ways can a person place 5 undistinguishable 6-side dice into 4 different containers? [56] • The sales manager of a clothing company needs to assign seven salespeople to seven different territories. How many possibilities are there for the assignments? [5040]

  25. What method to use? (cont) • In tossing 5 6-sided fair dice, what is the probability of at least one 2? • If the dice are distinguishable? • If the dice are indistinguishable?

  26. What method to use? (cont) 17. You roll 6 dice where the order is important, the first two are 4-sided, the next two are 6-sided and the last two are 10 sided. • How many different combinations are possible? [57,600] • What is the total number of combinations if no number can occur twice on the same size die? [32,400] • What is the total number of combinations if no number can occur twice on any size die? [4320]

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