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Counting Principles and Probability Concepts Explored

quinn-juarez avatar
quinn-juarez
25 diapositives · 360 Views · Nov 18, 2014

Dive into the Counting Principle, permutations, combinations, and probability theory with examples and explanations. Understand independent and dependent events, odds, and probability distributions. Learn how to apply these concepts to real-life scenarios.

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Counting Principles and Probability Concepts Explored
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  1. Chapter 12 Sec 1 The Counting Principle

  2. Independent Events • An outcome is the result of a single trial. • Flipping a coin. • The set of all possible outcomes is the sample set. • An event consists of one or more outcomes. • The choices of letters on a license plate are called independent events because each letter does not affect the choices for the others.

  3. Example 1 • A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayo. How many different combinations of bread and spread are possible. Method 1: Tree Bread W Wh R Spread B Mu Ma B Mu Ma B Mu Ma Possible WB WMuWMaWhBWhMuWhMa RB RMuRMa There are 9 possible outcomes

  4. The Fundamental Counting Principle

  5. Example 2 Many answering machines allow owners to call home and get their messages by entering a 3-digit code. How many codes are possible? 10 x 10 x 10 = 100 If the code is just 2 digits? 10 x 10 = 100

  6. Dependent Events • With dependent events the outcome of one event does affect the outcome of another. • The fundamental Counting principle applies to dependent events as well.

  7. Example 1 • Carlin wants to take 6 different classes next year. Assume that each class is offered each period, how many different schedules could he have? • When he schedules a class that class won’t be available to for the following periods. So, • Per 1 – 6 Choices Per 2 – 5 Per 3 – 4 … Per 6 – 1 • 6 x 5 x 4 x 3 x 2 x 1 or 6! = • 720 Choices

  8. Sum it up (get it, counting principles…sum…never mind)

  9. Chapter 12 Sec 2 Permutations and Combinations

  10. Permutations • When a group of objects or people are arranged in a certain order, the arrangement is called a permutation. In permutation, the order of the objects is very important • The arrangement in a line is call linear permutation.

  11. Permutation • Notice that is the first 4 factors of 7!. • You can rewrite this in terms of 7!.

  12. Example 1 Eight people enter the Best Pie contest. How many ways can blue, red and green ribbons be awarded. n = 8 and r = 3 P(n, r) =

  13. Permutation with Repetitions • How many different ways can the letters of the word BANANA be arranged? • There are 2 Ns and 3 As.

  14. Combinations • An arrangement of objects in which order is not important is called a combination. • The number of combinations of n object take r at a time is written C(n, r) or nCr..

  15. Example 2 Five cousins at a family reunion decide that three of them will go pick up a pizza. How many ways can they choose the three to go?

  16. Example 3 Six cards are drawn from a standard deck of cards. How many hands consist of two hearts and four spades? Hearts - C(13,2) Spades - C(13,4)

  17. Chapter 12 Sec 3 Probability

  18. Probability • The probability of an event is a ratio that measures the chances of the event occurring. • A desired outcome is called a success. • Any other outcome is called a failure

  19. The Proverbial Coin Toss When three coins are tossed, what is the probability that all three will be heads? First Coin H T 2nd Coin H T H T And 3rd H T H T H T H T Possible outcomes HHH HHT HTH HTT THH THT TTH TTT 8 Possible 1 Success and 7 Failure

  20. Example 2 Roman has a collection of 26 books – 16 are fiction and 10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability that he will chooses 4 fiction and 4 nonfiction? First determine how many ways you can get 4 of each. To find s, use the Fundamental Counting Principle 1820 x 210 = 382,200 Total of s + f SO…

  21. Odds • Another way t measure chance is with odds. • The odds that an event will occur can be expressed as a ratio of the number of successes to the number of failures.

  22. Example 3 According to the US National Center for Health Stats, the chances of a male born in 1990 living to be at least 65 years of age are about 3 in 4. For females, the chances are about 17 in 20. a. What are the odds of a male living to be at least 65? 3 out of 4 males will make it so successes = 3 4 – 3 will equal failures = 1 odds are 3:1 b. What are the odds of a female living to be at least 65? 17 out of 20 females will make it so successes = 17 20 – 17 will equal failures = 3 odds are 17:3

  23. Probability Distribution • Many experiments have numerical results. • A random variable is the numerical outcome of a random event. • A probability distribution for a random variable is a function that maps the sample space to the probabilities of the outcomes. • ie The table below illustrates the probability distribution for casting a die.

  24. Example 4 b. Use the table to find P(S = 9). What other sum has the same probability. c. What are the odds of rolling a sum of 7? a. Use the graph to determine which outcome is most likely What is the probability? • The greatest probability is and the most likely outcome is a sum of 7. • The P(9) is and the other outcome is 5. c. Identify s and f. Odds: s:f or 1:5 Suppose two dice are rolled. The table and the relative-frequency histogram show the distribution of the sum of the numbers rolled.

  25. Daily Assignment • Chapter 12 Sections 1 – 3 • Study Guide (SG) • Pg 157 – 162 All