TFG0152&TFL0152 Tölfræði

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TFG0152&TFL0152 Tölfræði Fyrirlestur 7 Kafli 4.5-4.

1 TFG0152&TFL0152 Tölfræði Fyrirlestur 7 Kafli Probability/Líkindi theoretically/fræðilega The probability of heads P(H) = ½ = 0,5. The probability of tails P(T) = ½ = 0,5. mutually exclusive events ósamrýmanlegir atburðir 3/2/2005 (c) Thomson Learning, Inc 1

2 event/atburður Compound Events The probability that either event A or event B will occur: P(A or B). P(heads or tails) = 1 The probability that both events A and B will occur: P(A and B). P(heads and tails) = 0 The probability that event A will occur given that event B has occurred: P(A B). Math2260 Venn diagram(addition Rule) diagram/skýringarmynd 3/2/2005 (c) Thomson Learning, Inc 2

3 Dæmi Visitors at an amusement park: 24 kl. eða 12 kl. All-Day Half-Day Pass Pass Total Male Female Total One visitor is selected at random: 1) The event A is the visitor purchased an all-day pass. 2) The event B as the visitor selected purchased a half-day pass. 3) The event C as the visitor selected is female. 3/2/2005 (c) Thomson Learning, Inc 3

4 Dæmi mutually exclusive events ósamrýmanlegir atburðir The events A and B are mutually exclusive. The events A and C are not mutually exclusive. The intersection of A and C can be seen in the table or in the Venn diagram below: A C /2/2005 (c) Thomson Learning, Inc 4

5 General Addition Rule Let A and B be two events defined in a sample space S: P( A or B) = P( A) + P( B) P( A and B) A B If two events A and B are mutually exclusive : P(A and B) = 0. 3/2/2005 (c) Thomson Learning, Inc 5

6 Special Addition Rule Let A and B be two events defined in a sample space. If A and B are mutually exclusive events, then: P( A or B) = P( A) + P( B) This can be expanded to consider more than two mutually exclusive events: P(A or B or C or D) = P(A) + P(B) + P(C) + P(D) A B C 3/2/2005 (c) Thomson Learning, Inc 6 D

7 complementary event/andstæður atburður Special Addition Rule Dæmi mutually exclusive events ósamrýmanlegir atburðir Workers at a hospital are classified as only one of the following: manager (A), doctor (B), nurse (C), or support (D). It is known that P(A) = 0.15, P(B) = 0.40, P(C) = 0.25, and P(D) = 0.20 P( A) = 1 P( A) = = 0.85 P( A and B ) = 0 (A and B are mutually exclusive ) P( B or C) = P( B) + P( C) = = 0.65 P( A or B or C) = P( A) + P( B) + P( C) = = /2/2005 (c) Thomson Learning, Inc 7

8 General Addition Rule Dæmi A consumer is selected at random. The probability the consumer has tried a snack food (F) is 0.5, tried a new soft drink (D) is 0.6, and tried both the snack food and the soft drink is 0.2 P( Tried the snack food or the soft drink) = P( F or D) = P( F) + P( D) P( F and D) = = 0.9 P( Not tried the snack food) = P( F) = 1 P( F) = = 0.5 P( Tried neither the snack food nor the soft drink) = P[( F or D)] = 1 P( F or D) = = 0.1 P( Tried only the soft drink ) = P( D) P( F and D ) = = 0.4 3/2/2005 (c) Thomson Learning, Inc 8

9 independent/óháður conditional probability/skilyrt líkindi Conditional Probability The symbol P(A B) represents the probability that A will occur given B has occurred. This is conditional probability. P( A B) = P( A and B) P( B) Given B has occurred, the relevant sample space is no longer S, but B (reduced sample space). Independent Events: Two events A and B are independent events if: P(A B) = P(A) or P(B A) = P(B) 3/2/2005 (c) Thomson Learning, Inc 9

10 jafn Dæmi events/atburðir even/jafn odd/ójafn Consider the experiment in which a single fair die is rolled: S = {1, 2, 3, 4, 5, 6 }. Define the following events: A = a 1 occurs, B = an odd number occurs, and C = an even number occurs. P( A B) P( A C) = P( A and B) / P( B) = / 6 = = P( A and C) P( C) = 0 3 / 6 = P( B A) P( B and A) 1/ 6 = = = P( A) 1/ 6 1 3/2/2005 (c) Thomson Learning, Inc 10

11 General Multiplication Rule Let A and B be two events defined in sample space S. Then: P( A and B) = P( A) P( BA ) or P( A and B) = P( B) P( A B) Special Multiplication Rule Let A and B be two events defined in sample space S. If A and B are independent events, then: P( A and B) = P( A) P( B) P(A and B and C and...) = P(A).P(B).P(C)... 3/2/2005 (c) Thomson Learning, Inc 11 HH HT TT TH P(H and H) = ¼

12 Special Multiplication Rule cold/kvef Dæmi Suppose the event A is Allen gets a cold this winter, event B is Bob gets a cold this winter, and event C is Chris gets a cold this winter. P(A) = 0.15, P(B) = 0.25, P(C) = 0.3, and all three events are independent. P( All three get colds this winter) = P( A and B and C) = P( A) P( B) P( C) = (0.15)(0.25)(0.30) = P( Allen and Bob get a cold, but Chris does not) = P( A and B and C) = P( A) P( B) P( C) = (0.15)(0.25)(0.70) = /2/2005 (c) Thomson Learning, Inc 12

13 tree/tré disease/sjúkdómur positive/jákvæður negative/neikvæður Tree Diagram - Dæmi Suppose the probability of having a disease (D) is If a person has the disease, the probability of a positive test result (Pos) is If a person does not have the disease, the probability of a negative test result (Neg) is For a person selected at random: a) Find the probability of a negative test result given the person has the disease. b) Find the probability of having the disease and a positive test result. c) Find the probability of a positive test result. 3/2/2005 (c) Thomson Learning, Inc 13

14 Tree Diagram Disease Test Result 0.90 Pos D 0.10 Neg En, þú ert með krabbamein! D 0.05 Pos En, þú ert ekki með krabbamein! complementary event/andstæður atburður 0.95 Neg 3/2/2005 (c) Thomson Learning, Inc 14

15 Svör a) P( Neg D) = 1 P( Pos D) = = 0.10 b) P( D and Pos) = P( D) P( Pos D) = (0.001)(0.90) = c) P( Pos) = P( D and Pos) + P( D and Pos) = P( D) P( Pos D) + P( D) P( Pos D) = (0.001)(0.90) +(0.999)(0.05) = /2/2005 (c) Thomson Learning, Inc 15

reliability/áreiðanleiki Space shuttle Challenger disaster Reliability of a single joint was 0.9777. But the reliability of six joints was: 0.9777 x 0.

16 reliability/áreiðanleiki Space shuttle Challenger disaster Reliability of a single joint was But the reliability of six joints was: x x x x 0,9777 x eða u.þ.b 87% Special Multiplication Rule Video: Chapter 4 and 5 probability 3/2/2005 (c) Thomson Learning, Inc 16

2011 Pearson Education, Inc

2011 Pearson Education, Inc Statistics for Business and Economics Chapter 3 Probability Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive