November 8, Chapter 8: Probability: The Mathematics of Chance

Transcription

1 Chapter 8: Probability: The Mathematics of Chance November 8, 2013

2 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes

3 Crystallographic notation The first symbol is always a p, which indicates that the pattern repeats (is periodic ) in the horizontal direction. The second symbol is m if there is a vertical line of reflection. Otherwise, it is 1. The third symbol is m (for mirror ), if there is a horizontal line of reflection (in which case there is also glide reflection) a (for alternating ), if there is a glide reflection but no horizontal reflection 1 if there is no horizontal reflection or glide reflection The fourth symbol is 2, if there is half-turn rotational symmetry; otherwise, it is 1.

4 Group A group is a collection of elements {A, B, } and an operation between pairs of them such that the following properties hold: Closure: The result of one element operating on another is itself an element of the collection (A B is in the collection). Identity element: There is a special element I, called the identity element, such that the result of an operation involving the identity and any element is that same element (I A = A and A I = A). Inverses: For any element A, there is another element, called its inverse and denoted A 1, such that the result of an operation involving an element and its inverse is the identity element (A A 1 = I and A 1 A = I ). Associativity: The result of several consecutive operations is the same regardless of grouping or parenthesizing, provided that the consecutive order of operations is maintained: A B C = A (B C) = (A B) C.

5 Probability Rules Probability Rules Rule 1. The probability P(A) of any event A satisfies 0 P(A) 1. Rule 2. If S is the sample space in a probability model, then P(S) = 1. Rule 3. The complement rule: P(A C ) = 1 P(A). Rule 4. The multiplication rule for independent events: P(A and B) = P(A) P(B). Rule 5. The general addition rule: P(A or B) = P(A) + P(B) P(A and B). Rule 6. The addition rule for disjoint events: P(A or B) = P(A) + P(B).

6 Counting distinct items Counting Ordered Collections of Distinct Items Rule A. Suppose we have a collection of n distinct items. We want to arrange k of these items in order, and the same item can appear more than once in the arrangement. The number of possible arrangements is n n n = n k Rule B. (Permutations) Suppose we have a collection of n distinct items. We want to arrange k of these items in order, and any item can appear no more than once in the arrangement. The number of possible arrangements is n (n 1) (n k + 1)

7 Counting Distinct Items Counting Unordered Collections of Distinct Items Rule C. Suppose that we have a collection of n distinct items. We want to select k of those items with no regard to order, and any item can appear more than once in the collection. The number of possible collections is (n + k 1)! k!(n 1)! Rule D. (Combinations) Suppose that we have a collection of n distinct items. We want to select k of these items with no regard to order, and any item can appear no more than once in the collection. The number of possible selections is n! k!(n k)!

8 Question Choose a young adult (aged 25 to 34 years) at random. The probability is 0.12 that the person choose did not complete high school, 0.31 that the person has a high school diploma but no further education, and 0.29 that the person has at least a bachelor s degree. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor s degree? Answer: =.28 28% (b) What is the probability that a randomly chosen young adult has at least a high school education? Answer: =.88 88%

9 Question You toss a balanced coin 10 times and write down the resulting sequence of heads and tails. (a) How many possible outcomes are there for 10 tosses? Answer: 2 10 (b) What is the probability that your 10-toss sequence is either all heads or all tails? Answer: = = (c) What is the probability that your 10-toss sequence has a combine 5 heads and a combine 5 tails? Answer: ( 10 )

10 Question In poker, a royal flush is a 5-card hand containing an ace, king, queen, jack, and 10, all of the same suit. (a) How many royal flush hands are possible? Answer: 4 (b) What is the number of 5-card hands possible from a 52-card deck? Answer: ( ) 52 5 (c) What is the probability that 5 cards drawn at random from a 52-card deck will yield a royal flush? Answer: 4 ( 52 5 )

11 Question Suppose a monkey is at a type writer and can only press a, r, e. 1 How many possible three-letter words can the monkey type using only these letters? Answer: 3 3 = 27 2 Which of these are words in an English dictionary? Answer: are, era, ear 3 What is the probability that the word the monkey typed is in a English dictionary? Answer:

12 Questions How many different ways can 3 dice sum to 8? How many different ways can 3 dice sum to 9? How many different ways can 3 dice sum to 13? How many different ways can 3 dice sum to 12? How many different ways can 4 dice sum to 9?

13 Questions How many different ways can 3 dice sum to 8?

14 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9?

15 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13?

16 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13? Answer: 21 How many different ways can 3 dice sum to 12?

17 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13? Answer: 21 How many different ways can 3 dice sum to 12? Answer: 25 How many different ways can 4 dice sum to 9?

18 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13? Answer: 21 How many different ways can 3 dice sum to 12? Answer: 25 How many different ways can 4 dice sum to 9? Answer: 56

19 Question A computer assigns three-character log-in IDs that may contain the digits 0 to 9 as well as the letters a to z, with repeats allowed. 1 What is the probability that your ID contains no x? 2 What is the probability that your ID contains no digits? 3 What is the probability that your ID contains exactly 2 x s?

20 Question What is the probability that two people in this room have the same birthday?

21 Question What is the probability that two people in this room have the same birthday? Answer: 1 365! (365 14)! = 0.223

22 Question How many different ways can you seat 10 couples at a circular table such that everyone is sitting next to their spouse? Assuming we alternate men and women how arrangements have at least one couple not sitting together?

23 Next time Continuous Probability Models