advertisement

Assignment 3.1 due Tuesday March 4, 2014 Sta220 - Statistics Mr. Smith Room 310 Class #8 Section 3.2-3.4 3.2 Unions and Intersections • The union of two events A and B is the event that occurs if either A or B (or both) occurs on a single performance of the experiment. We denote the union of events A and B by the symbol A U B. • The intersection of two events A and B is the event that occurs if both A and B occur on a single performance of the experiment. We write ∩ for the intersection of A and B. Venn diagrams for union and intersection Copyright © 2013 Pearson Education, Inc.. All rights reserved. Problem: Consider a die-toss experiment in which the following events are defined: A: {Toss an even number.} B: {Toss a number less than or equal to 3.} a) Describe ∪ for this experiment. b) Describe ∩ for this experiment. c) Calculate ( ∪ ) and P( ∩ ),assuming that the die is fair. Figure 3.8 Venn diagrams for die toss Copyright © 2013 Pearson Education, Inc.. All rights reserved. Solution a. ∪ = {1, 2, 3, 4, 6} b. ∩ = 2 c. ∪ = 1 + 2 + 3 + 4 + 6 = ∩ = 1 6 1 1 1 1 + + + + 6 6 6 6 1 2 = 6 = 5 6 Note: Unions and intersections can be defined for more than two events. For example, the event A U B U C represents the union of three events: A, B, and C. This event, which includes the set of sample points in A, B, or C, will occur if any one (or more) of the events A, B, and C occurs. Similarly, the intersection ∩ ∩ is the event that all three of the events A, B, and C occur. Problem: Family Planning Perspectives reported on a study of over 200,000 births in New Jersey over a recent two-year period. The study investigated the link between the mother’s race and the age at which she gave birth (called maternal age). The percentages of the total number of births in New Jersey, by the maternal age and race classifications, are given in Table 3.4. Table 3.4 Percentage of New Jersey Birth Mothers, by Age and Race Race Maternal Age (Years) White White Black ≤ 2% 2% 18-19 3% 2% 20-29 41% 12% ≥ 33% 5% This table is called a two-way table, since responses are classified according to two variables: maternal age (rows) and race (columns). Define the following event: A: {A New Jersey birth mother is white.} B: {A New Jersey mother was a teenager when giving birth.} a) Find P(A) and P(B). b) Find P( ∪ ) . c) Find ( ∩ ). First Identify you sample points 1 : {≤ 17, ℎ} 2 : {18 − 19, ℎ} 3 : {20 − 29, ℎ} 4 : {≥ 30 , ℎ 5 : {≤ 17, } 6 : {18 − 19 , } 7 : {20 − 29, } 8 : {≥ 30, } (1 ) = ℎ ℎ − {≤ 17, ℎ}= .02 (2 ) =.03 (3 ) =.41 (4 ) =.33 (5 ) =.02 (6 ) =.02 (7 ) =.12 (8 ) =.05 Solution On the White Board 3.3 Complementary Events The complement of an event A is the event that A does not occur -that is, the event consisting of all sample points that are not in event A. We denote the complement of A by RULE of COMPLEMENTS An event A is a collection of sample points, and the sample points included in are those not in A. () + ( ) = 1 Problem: Consider the experiment of tossing fair coins. Define the following event: A: {Observing at least one head}. a) Find P(A) if 2 coins are tossed. b) Find P(A) if 10 coins are tossed. Solution On the White Board 3.4 The Additive Rule and Mutually Exclusive Events Additive Rule of Probability The probability of the union of events A and B is the sum of the probability of event A and the probability of event B, minus the probability of the intersection of events A and B; that is ∪ = + − ( ∩ ) Problem Hospital records show that 12% of all patients are admitted for surgical treatment, 16% are admitted for obstetrics, and 2% receive both obstetrics and surgical treatment. If a new patient is admitted to the hospital, what is the probability that the patient will be admitted for surgery, for obstetrics, or for both? Solution Consider the following events: A: {A patient admitted to the hospital receives surgical treatment} B:{A patient admitted to the hospital receives obstetrics treatment} Solution P(A) = .12 P(B) = .16 ∩ = .02 What is ∪ = ? Events A and B are mutually exclusive if ∩ contains no sample points – that is, if A and B have no sample points in common. For mutually exclusive events, ∩ =0 Probability of Union of Two Mutually Exclusive Events If two events A and B are mutually exclusive, the probability of the union of A and B equals the sum of the probability of A and the probability of B; that is ∪ = + (). CAUTION: The preceding formula is false if the events are not mutually exclusive. In that case (i.e. two non-mutually exclusive events), you must apply the general additive rule of probability. Problem Consider the experiment of tossing two balanced coins. Find the probability of observing at least one head. Solution Define the events A: {Observe at least one head} B: {Observe exactly one head} C: {Observe exactly two heads} Note A = ∪ Assignment 3.2-3.4 due Friday March 7, 2014 at the beginning of class. This assignment is worth 45 points. Make sure you read the directions.