CHAPTER 4 4 2 Addition Rules for Probability

CHAPTER 4 4 -2 Addition Rules for Probability Instructor: Alaa saud Note: This Power. Point is only a summary and your main source should be the book.

q Two events are mutually exclusive events if they cannot occur at the same time (i. e. , they have no outcomes in common) Addition Rule P(A or B)=P(A)+P(B) Mutually Exclusive q This means that P(A∩B)= 0 i. e. the two event cannot occur at the same time. P (S) B A P(A or B)=P(A)+P(B)- P(A and B) Not Mutually Exclusive q Where P(A∩B) is the probability both A and B occur. P(A∩B) P (S) B A Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -15: Determining whether the two events are mutually exclusive. a. Randomly selecting a female student. Randomly selecting a student who is a junior These event are not mutually exclusive b. Randomly selecting a person with type A blood. Randomly selecting a person with type B blood These event are mutually exclusive b. Rolling a die and getting an odd number. Rolling a die and getting a number less than 3. These event are not mutually exclusive d. Randomly selecting a person who is under 21 years of age. Randomly selecting a person who is over 30 years of age. These event are mutually exclusive

Example 4 -17: A city has 9 coffee shops: 3 Starbucks, 2 Caribou Coffees, and 4 Crazy Mocho Coffees. If person selects one shope at random to buy a cup of coffee, Find the probability that it is either a Starbucks or Crazy Mocho Coffees Solution : P(Starbucks or Crazy Mocho )= P(Starbucks) + P(Crazy Mocho ) The events are mutually exclusive

Example 4 -17: In a survey, 8%of the respondent said that their favorite ice cream flavor is cookies and cream, and 6%like mint chocolate chip. If person selected at random Find the probability that her favorite ice cream flavor is either cookies and cream or mint chocolate chips. Solution : P(cookies and cream or mint chocolate chips) =P(cookies and cream ) + P(mint chocolate chips) The events are mutually exclusive

Example 4 -21: In a hospital unit there are 8 nurses and 5 physicians ; 7 nurses and 3 physicians are females. If a staff person is selected , find the probability that the subject is a nurse or a male. Solution : The sample space Staff Females Males Total Nurses 7 1 8 Physicians 3 2 5 Total 10 3 13 Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -22: The probability of person driving while intoxicated is 0. 32, the probability of a person having a driving accident is 0. 09, and the probability of a person having driving accident while intoxicated is 0. 06. What is the probability of a person driving while intoxicated or having driving accident? Solution : P(intoxicated or accident)= P(intoxicated) + P(accident)-P(intoxicated and accident) =0. 32+0. 09 -0. 06=0. 35 The events are not mutually exclusive

• Which one of these events is not mutually exclusive? A) Select a student in your university: The student is married, and the student is a business major. • B) Select a ball from bag: It is a football, and it is a basket ball. • C) Roll a die: Get an even number, and get an odd number. • D) Select any course: It is an Arabic course, and it is a Statistics course.

Determine which events are mutually exclusive. a) Select a student in your college: The student is in the second year and the student is a math major. b) Select a child: The child has black hair and the child has black eyes. c) Roll a die: Get a number greater than 2 and get a multiple of 3. d) Roll a die: Get a number greater than 3 and get a number less than 3.

If P(A) = 0. 3, P(B) = 0. 4 , and A, B are mutually exclusive events, find P(A and B). a) 0 b) 1 c) 0. 12 d) 0. 7

4 -3 The Multiplication Rules and Conditional Probability Note: This Power. Point is only a summary and your main source should be the book.

q Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring. Multiplication Rules P(A and B)=P(A). P(B) Independent P(A and B)=P(A). P(B|A) Dependent Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -23: A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die Solution :

Example 4 -25: An urn contains 3 red balls , 2 blue balls and 5 white balls. A ball is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of each of these. a. Selecting 2 blue balls Note: This Power. Point is only a summary and your main source should be the book.

b. Selecting 1 blue ball and then 1 white ball c. Selecting 1 red ball and then 1 blue ball Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -27: Approximately 9% of men have a type of color blindness that prevents them from distinguishing between red and green. If 3 men are selected at random , find the probability that all of them will have this type of red-green color blindness. Solution : Let C denote red – green color blindness. Then P(C and C) = P(C) = (0. 09)(0. 09) = 0. 000729 Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -28: In a rcent survey, 33% of the respondent said that they feel that they are overqualified (O) for their present job. Of these, 24% said that they were looking for a new job (J). If a person is selected at random, find the probability that the person feels that he is overqualified and is also looking for a new job Solution :

Example 4 -29: World Wide Insurance Company found that 53% of the residents of a city had homeowner’s insurance (H) with the company. Of these clients , 27% also had automobile insurance (A) with the company. If a resident is selected at random , find the probability that the resident has both homeowner’s and automobile insurance with World Wide Insurance Company. Solution : Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -31: Box 1 contains 2 red balls and 1 blue ball. Box 2 contains 3 blue balls and 1 red ball. A coin is tossed. If it falls heads up , box 1 is selected and a ball is drawn. If it falls tails up , box 2 is selected and a ball is drawn. Find the probability of selecting a red ball. Box 1 Box 2 Note: This Power. Point is only a summary and your main source should be the book.

Solution : Red Box 1 Blue Coin Red Box 2 Blue Note: This Power. Point is only a summary and your main source should be the book.

Q(1): box 1 contains 20% defective transistors, box 2 contains 30% defective transistors, and box 3 contains 50% defective transistors. A die is rolled. If the number that appears is greater than 3, a transistor is selected from 1. If the number is less than 3, a transistor is selected from 2. If the number is 3, a transistor is selected from. Find the probability of selecting a defective transistor. a) 0. 028 b) 1 c) 0. 283 d) 0. 03 Q(2): A die is rolled. What is the probability that the number rolled is greater than 2 and even number? 1/3

Q(3): Box A contains 4 red balls and 2 white balls. Box B contains 2 red balls, 2 white balls. A die is rolled first and if the outcome is an even number a ball is chosen at random from Box A, and if the outcome is an odd number a ball is randomly chosen from Box B. Find the probability that a red ball is chosen?

Conditional Probability Note: This Power. Point is only a summary and your main source should be the book.

Conditional probability qis the probability that the second event B occurs given that the first event A has occurred. Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -32: A box contains black chips and white chips. A person selects two chips without replacement. If the probability of selecting a black chip and a white chip is , and the probability of selecting a black chip on the first draw is , find the probability of selecting the white chip on the second draw , given that the first chip selected was a black chip. Note: This Power. Point is only a summary and your main source should be the book.

Solution : Let B=selecting a black chip W=selecting a white chip Hence , the probability of selecting a while chip on the second draw given that the first chip selected was black is Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -34: Survey on Women In the Military A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown.

Solution : a. Find the probability that the respondent answered yes (Y), given that the respondent was a female (F).

b. Find the probability that the respondent was a male (M), given that the respondent answered no (N).

Probabilities for “At Least” • A coin is tossed 3 times. Find the probability of getting at least 1 tail ? Note: This Power. Point is only a summary and your main source should be the book.

Example 4 -36: • A coin is tossed 5 times. Find the probability of getting at least 1 tail ? E=at least 1 tail E= no tail ( all heads) P(E)=1 -P(E) P(at least 1 tail)=1 - p(all heads) Note: This Power. Point is only a summary and your main source should be the book.

For Example: It has been found that 8% of all automobiles on the road have defective brakes. If 8 automobiles are stopped and checked by the police , find the probability that at least one will have defective brakes. Solution : P(at least one will have defective brakes) = 1 – p( all have not defective brakes) = 1 - (1 - 0. 08)8 = 1 - (0. 92)8 = 0. 487

1)It is reported that 27% of working women use computers at work. Choose 5 working women at random. A. Find the probability that at least 1 use a computer at work. B. Find the probability that at least 1 doesn’t use a computer at work. C. Find the probability that all 5 use a computer in their jobs. 2)Only 27% students take computer course. Choose 3 students at random. Find the probability that A. All 3 take computer course B. All 3 dose not take computer course C. At least 1 of three dose not take computer course?

3) 70% of females have never been married choose 5 women find : A. The probability non have ever been married ? B. The probability all have been married ? C. At least 1 has been married ? D. At least 1 has been not married? 4) The probabilities of the events A and B are: P(A and B)=0. 2, P(B│A)=0. 3 Find the ?