Basic Concepts of Probability and Counting Section 3.1

Transcription

1 Basic Concepts of Probability and Counting Section 3.1 Summer Math 1040 June 17 (1040) M June 17 1 / 12

2 Roadmap Basic Concepts of Probability and Counting Pages Counting events, and The Fundamental Counting Principle Theoretical probability and statistical probability This section introduces the concept of a sample space, a list of all possible outcomes of a probability experiment. Counting these events allow us to find the probability of an event. (1040) M June 17 2 / 12

3 Sample Spaces A sample space develops by listing all possible results from a random experiment. (1040) M June 17 3 / 12

4 Sample Spaces A sample space develops by listing all possible results from a random experiment. Example Rolling a 4-sided die s sample space is {1, 2, 3, 4}. Example A coin flip s outcome is {H, T } for heads and tails. Example Possible answer s to, Do you want kids? is a sample space: {Yes, No, Maybe}. (1040) M June 17 3 / 12

5 Events Particular outcomes is called an event. Example: We roll a 4-sided die. Here are some possible events: You roll less than a 4. (1040) M June 17 4 / 12

6 Events Particular outcomes is called an event. Example: We roll a 4-sided die. Here are some possible events: You roll less than a 4. {1, 2, 3} There are 3 ways. (1040) M June 17 4 / 12

7 Events Particular outcomes is called an event. Example: We roll a 4-sided die. Here are some possible events: You roll less than a 4. {1, 2, 3} There are 3 ways. You roll an odd number. (1040) M June 17 4 / 12

8 Events Particular outcomes is called an event. Example: We roll a 4-sided die. Here are some possible events: You roll less than a 4. {1, 2, 3} There are 3 ways. You roll an odd number. {1, 3} There are 2 ways. (1040) M June 17 4 / 12

9 Events Particular outcomes is called an event. Example: We roll a 4-sided die. Here are some possible events: You roll less than a 4. {1, 2, 3} There are 3 ways. You roll an odd number. {1, 3} There are 2 ways. You roll a prime number. (1040) M June 17 4 / 12

10 Events Particular outcomes is called an event. Example: We roll a 4-sided die. Here are some possible events: You roll less than a 4. {1, 2, 3} There are 3 ways. You roll an odd number. {1, 3} There are 2 ways. You roll a prime number. {2, 3} There are 2 ways. (1040) M June 17 4 / 12

11 Fundamental Counting Principle If we combine two (or more) basic types of experiments, counting the possible number of outcomes is found by multiplying the number of outcomes in each sample space. Example Rolling a 4-sided die and flipping a coin s sample space has 4 2 = 8 outcomes: {1H, 2H, 3H, 4H, 1T, 2T, 3T, 4T } (1040) M June 17 5 / 12

12 Fundamental Counting Principle If we combine two (or more) basic types of experiments, counting the possible number of outcomes is found by multiplying the number of outcomes in each sample space. Example Rolling a 4-sided die and flipping a coin s sample space has 4 2 = 8 outcomes: {1H, 2H, 3H, 4H, 1T, 2T, 3T, 4T } For an event, the rule is the same. Multiply the number of ways to do the first event with the number of ways to do the next event. (1040) M June 17 5 / 12

13 Fundamental Counting Principle Example A restaurant offers four different main dishes and 3 different desserts. If a meal comes with a main dish and a dessert, how many different means can be made? (1040) M June 17 6 / 12

14 Fundamental Counting Principle Example A restaurant offers four different main dishes and 3 different desserts. If a meal comes with a main dish and a dessert, how many different means can be made? Answer 4 3 = 12 many meals. (1040) M June 17 6 / 12

15 Fundamental Counting Principle Example A restaurant offers four different main dishes and 3 different desserts. If a meal comes with a main dish and a dessert, how many different means can be made? Answer 4 3 = 12 many meals. Example How many 4-character liceanse plates can be made from 26 letters and 10 digits (zero through nine)? (1040) M June 17 6 / 12

16 Fundamental Counting Principle Example A restaurant offers four different main dishes and 3 different desserts. If a meal comes with a main dish and a dessert, how many different means can be made? Answer 4 3 = 12 many meals. Example How many 4-character liceanse plates can be made from 26 letters and 10 digits (zero through nine)? Answer There are 36 different characters each time = 36 4 = 1, 679, 616 many ways. (1040) M June 17 6 / 12

17 Fundamental Counting Principle Example A restaurant offers four different main dishes and 3 different desserts. If a meal comes with a main dish and a dessert, how many different means can be made? Answer 4 3 = 12 many meals. Example How many 4-character liceanse plates can be made from 26 letters and 10 digits (zero through nine)? Answer There are 36 different characters each time = 36 4 = 1, 679, 616 many ways. This is the fundamental counting principle: The number of ways two events can occur in sequence is m n, the product of the number of ways m the first and the number of ways n the second can occur. This extends to more than two events. (1040) M June 17 6 / 12

18 Classical / Theoretical Probability The probability an event E will occur is denoted P(E) and said, the probability of event E. Classical or theoretical probability is used when each outcome in a sample space is equally likely to occur. The probability of an event E is then Number of outcomes in E P(E) = Total outcomes in the sample space (1040) M June 17 7 / 12

19 Classical / Theoretical Probability The probability an event E will occur is denoted P(E) and said, the probability of event E. Classical or theoretical probability is used when each outcome in a sample space is equally likely to occur. The probability of an event E is then Number of outcomes in E P(E) = Total outcomes in the sample space Example For a coin flip, the sample space is {H, T }. The event E : coin flip results in a heads is 1 2. (1040) M June 17 7 / 12

20 Classical / Theoretical Probability Example A card is drawn from a standard deck of playing cards. What is the probability that the card drawn is a heart? (1040) M June 17 8 / 12

21 Classical / Theoretical Probability Example A card is drawn from a standard deck of playing cards. What is the probability that the card drawn is a heart? P(E) = = 1 4 = (1040) M June 17 8 / 12

22 Classical / Theoretical Probability Example A card is drawn from a standard deck of playing cards. What is the probability that the card drawn is a heart? P(E) = = 1 4 = What is the probability the card is a face card? (A jack, queen, king, or ace) (1040) M June 17 8 / 12

23 Classical / Theoretical Probability Example A card is drawn from a standard deck of playing cards. What is the probability that the card drawn is a heart? P(E) = = 1 4 = What is the probability the card is a face card? (A jack, queen, king, or ace) There are four suits (heart, diamond, club, spade) and four face cards. P(E) = = (1040) M June 17 8 / 12

24 Empirical / Statistical Probability Empirical or statistical probabilities are based on observations. These are always relative frequencies. P(E) = f n = Frequency of the event Frequency total (1040) M June 17 9 / 12

25 Classical / Theoretical Probability Example Here is the toy dog breed data from the American Kennel Society (registered number of dogs in thousands) Chihuahua 23 Maltese 13 Pomeranian 18 Poodle 30 Pug 20 Shih Tzu 27 Yorkshire Terrier 48 Σf = 179 What is the probability the next dog registered is a poodle? (1040) M June / 12

26 Classical / Theoretical Probability Example Here is the toy dog breed data from the American Kennel Society (registered number of dogs in thousands) Chihuahua 23 Maltese 13 Pomeranian 18 Poodle 30 Pug 20 Shih Tzu 27 Yorkshire Terrier 48 Σf = 179 What is the probability the next dog registered is a poodle? P(E) = (1040) M June / 12

27 Complementary Events Because probabiity must be a number between 0 and 1, we can use this fact to find the probabiliy of the complement of E, or all the events not in E. This is done by P(E ) = 1 P(E) (1040) M June / 12

28 Complementary Events Because probabiity must be a number between 0 and 1, we can use this fact to find the probabiliy of the complement of E, or all the events not in E. This is done by P(E ) = 1 P(E) Example What is the probability that a card drawn from a standard deck is not a heart? (1040) M June / 12

29 Complementary Events Because probabiity must be a number between 0 and 1, we can use this fact to find the probabiliy of the complement of E, or all the events not in E. This is done by P(E ) = 1 P(E) Example What is the probability that a card drawn from a standard deck is not a heart? Let E be the card is a heart. (1040) M June / 12

30 Complementary Events Because probabiity must be a number between 0 and 1, we can use this fact to find the probabiliy of the complement of E, or all the events not in E. This is done by P(E ) = 1 P(E) Example What is the probability that a card drawn from a standard deck is not a heart? Let E be the card is a heart. P(E ) = 1 P(E) = = (1040) M June / 12

31 Complementary Events Because probabiity must be a number between 0 and 1, we can use this fact to find the probabiliy of the complement of E, or all the events not in E. This is done by P(E ) = 1 P(E) Example What is the probability that a card drawn from a standard deck is not a heart? Let E be the card is a heart. P(E ) = 1 P(E) = = What is the probabiliy that a card drawn is not a face card? (1040) M June / 12

32 Complementary Events Because probabiity must be a number between 0 and 1, we can use this fact to find the probabiliy of the complement of E, or all the events not in E. This is done by P(E ) = 1 P(E) Example What is the probability that a card drawn from a standard deck is not a heart? Let E be the card is a heart. P(E ) = 1 P(E) = = What is the probabiliy that a card drawn is not a face card? Let E be the card is a face card. (1040) M June / 12

33 Complementary Events Because probabiity must be a number between 0 and 1, we can use this fact to find the probabiliy of the complement of E, or all the events not in E. This is done by P(E ) = 1 P(E) Example What is the probability that a card drawn from a standard deck is not a heart? Let E be the card is a heart. P(E ) = 1 P(E) = = What is the probabiliy that a card drawn is not a face card? Let E be the card is a face card. P(E ) = 1 P(E) = (1040) M June / 12

34 Assignments Assignment: 1. Summarize this section. 2. Read pages Page 138, 1-73 odd 4. Try It Yourself exercises 1, 3, 4, 5, 7, 9 Vocabulary: sample space, event, the fundamental counting principle, theoretical probability, statistical probability, complementary events Understand: Write out a list of all possilbe outcomes of an experiment. This is the sample space. Count these events, and add up these events. This way you can compute probabilites. Use techniques such as the fundamental counting principle and the complement rule. (1040) M June / 12