Mutually Exclusive Events & Non-Mutually Exclusive Events. When two events A and B are mutually exclusive, the probability that A or B will occur is

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1 EVENTS & PROBABILITIES RULES PROBABILITY RULES Mutually Exclusive Events & Non-Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time (they have no outcomes in common). The probability of two or more events can be determined by the addition rules. There are two addition rules to determine either the two events are mutually exclusive or not mutually exclusive. Addition Rule 1 When two events A and B are mutually exclusive, the probability that A or B will occur is A or B) A) + B) or A and B) 0 A) B) Addition Rule When two events A and B are not mutually exclusive, then A or B) A) + B) A and B) A and B) A) B) Introduction to Probability Dr.Hayder Abbas Drebee 1

2 Consider the following events when rolling a die: A an even number is obtained,4, B an odd number is obtained 1,, Are events A and B are mutually exclusive? Solution: Yes, the two events are mutually exclusive since event A and event B have no common element, A B 4 1 Determine which events are mutually exclusive and which are not when a single die is rolled. a) Getting a and getting an odd number. Answer: Not Mutually Exclusive b) Getting a number greater than 4 and getting a number less than 4. Answer: Mutually Exclusive c) Getting an odd number and getting a number less than 4. Answer: Not Mutually Exclusive There are 8 nurses and physicians in a hospital unit; nurses and physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male. Solution: Staff Female, F Male, M Total Nurses, N 1 8 Physicians, PY Total 1 Introduction to Probability Dr.Hayder Abbas Drebee

3 P (N or M) N M) N) + M) N M) At a convention there are mathematics instructors, computer sciences instructors, statistics instructors, and 4 science instructors. If an instructor is selected, find the probability of getting a science instructor or a math instructor. Solution: P (science instructor or math instructor) P (S M) A grocery store employs cashiers, stock clerks and deli personnel. The distribution of employees according to marital status is shown here. Marital Status Cashiers Clerks Deli Personnel Married 8 1 Not Married 1 If an employee is selected at random, find these probabilities: a. the employee is a stock clerk or married clerk married) clerk) + married) - clerk married) b. the employee is not married 1 not married) c. the employee is a cashier or is unmarried cashier not married) cashier) + not married) cashier not married) Introduction to Probability Dr.Hayder Abbas Drebee

4 Independent & Dependent Events For two independent events, A and B, the occurrence of event A does not change the probability of B occurring. The probability of independent events can be determined as: A B ) A) Or B A ) B) Multiplication Rule 1 When two events are independent, the probability of both occurring A B) A) B) A box contains red balls, blue balls, and white balls. A ball is selected and its colour noted. Then it is replaced. A second ball is selected and its colour noted. Find the probability of each of these: a) selecting two blue balls. P (blueblue) blue) blue) b) selecting 1 blue ball and then 1 white ball. P (bluewhite) blue) white) 0 1 c) selecting 1 red ball and then 1 blue ball. redblue) red) blue) A survey found that 8% of book buyers are 40 years or older. If two book buyers are selected at random, find the probability that both are 40 years or older. Introduction to Probability Dr.Hayder Abbas Drebee 4

5 P (buyer) 0.8 x On the other hand, two events, A and B are dependent when the occurrence of the event A changes the probability of the occurrence of event B. When two events are dependent, another multiplication rule can be used to find the probability. Multiplication Rule When two events are dependent, the probability of both occurring P (A B) A) B A ) In a scientific study there are 8 tigresses, of which are pregnant. If are selected at random without replacement, find the probability that: a) all tigresses are pregnant. 1 st tigress nd tigress rd tigress Outcomes 8 8 ) ,,,,,,,,,,,,,,,, Introduction to Probability Dr.Hayder Abbas Drebee

6 b) two tigresses are pregnant. Let A be an event of two tigresses are pregnant A) ) + ) + ) ( ) ( ) ( ) Complementary Events The set of outcomes in the sample space that is not included in the outcomes of event E. Denoted as E (read E bar ) Find the complement of each event. a) Rolling a die and getting a 4 Answer: 1,,, and b) Selecting a letter of the alphabet and getting a vowel Answer: A Consonant c) Selecting a day of the week and getting a weekday Answer: Saturday or Sunday The outcomes of an event and the outcomes of the complement make up the entire sample space. The rule of complementary events can be stated algebraically in three ways: E) 1 E) Or E) 1 E) Or Introduction to Probability Dr.Hayder Abbas Drebee

7 P ( E) E) 1 The concept can be represented pictorially by the following Venn Diagram. E) E) S)1 P (E) In a group of 000 taxpayers, have been audited by the IRS at least once. If one taxpayer is randomly selected from this group, what are the probability of that taxpayer has never been audited by the IRS? Solution: Let, A the selected taxpayer has been audited by the IRS at least once A the selected taxpayer has never been audited by the IRS A) 1 A) 1 ( / 000) 1-1 / 4 / The multiplication rules can be used with the complementary event rule to simplify solving probability problems involving at least. In a department store there are customers, 90 of whom will buy at least one item. If 4 customers are selected at random, one by one, find the probability that at least one of the customers will but at least one item. Would you consider this event likely to occur? Explain. Solution: Let C at least one customer will buy at least one item C none of the customers will buy at least one item will buy at least one item) 90 / ¾ So, won t buy any items) 1 - /4 ¼ By using the complementary event rule, C) 1 C) Introduction to Probability Dr.Hayder Abbas Drebee

8 Yes, this event is most likely to occur (certain event) since the probability almost 1 NOTE: The following examples are based on the overall understanding of the entire probability concepts A random sample of college students was asked if college athletes should be paid. The following table gives a two-way classification of the responses. Should be paid, Should not be Total PAID paid, PAID Student athlete, SA 90 0 Student non-athlete, SNA Total 00 0 a) If one student is randomly selected from these students, find the probability that this student i. Is in favour of paying college athletes PAID) PAIDSA) + PAIDSNA) 90 4 ii. Favours paying college athletes given that the student selected is a nonathlete PAID SNA) PAID SNA) SNA) / 00 / iii. Is an athlete and favours paying student athletes SA PAID) iv. Is a non-athlete or is against paying student athletes SNA PAID ) SNA) + PAID ) - SNA PAID ) Introduction to Probability Dr.Hayder Abbas Drebee 8

9 1 40 b) Are the events student athlete and should be paid independent? Are they mutually exclusive? Explain why or why not. SAPAID) 9/ and SA) PAID) Since, SAPAID) SA) PAID), those two events are not independent (dependent). And since SAPAID) 0, those two events are not mutually exclusive A screening test for a certain disease is prone to giving false positives of false negatives. If a patient being tested has the disease, the probability that the test indicates a false negative is 0.1. If the patient does not have the disease, the probability that the test indicates a false positive is 0.. Assume that % of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that; Let D the patient has the disease D the patient does not have the disease PO the patient tests positive NE the patient tests negative 0.8 PO Joint Probability DPO) 0.0 D 0.1 NE DNE) 0.9 D PO NE D PO) D NE) Introduction to Probability Dr.Hayder Abbas Drebee 9

10 a) This patient has the disease and tests positive DPO) 0.0 x b) This patient does not have the disease and tests positive D PO) 0.9 x c) This patient tests positive PO) DPO) + D PO) d) This patient does not have the disease and tests negative D NE) 0.9 x e) This patient has the disease given that he/she tests positive D PO) D PO) PO) Introduction to Probability Dr.Hayder Abbas Drebee

11 Introduction to Probability Dr.Hayder Abbas Drebee 11