STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia

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1 STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that an event will happen can be described by a number between 0 and 1: A probability of 0, or 0%, means the event has no chance of happening. A probability of 1/2, or 50%, means the event is just as likely to happen as not to happen. A probability of 1, or 100%, means the event is certain to happen. For instance, the probability of a coin landing heads up is ½, or 50%, This means you would expect a coin to land heads up half of the time. 1

2 2.1 - Sample Space The sample space of a statistical experiment, denoted by S, is the set of all possible outcomes of that experiment. Ex. Roll a die Outcomes: landing with a 1, 2, 3, 4, 5, or 6 face up. Sample Space: S {1, 2, 3, 4, 5, 6} Sample Space Example 2 : A automobile consultant records fuel type and vehicle type for a sample of vehicles 2 Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV The Sample Space: e 1 e 1 e 2 e 3 e 4 e 5 e 6 Gasoline, Truck Gasoline, Car Gasoline, SUV Diesel, Truck Diesel, Car Diesel, SUV Car Car e 2 e 3 e 4 e 5 e 6 2

3 Sample Space Example 3: Suppose that three items are selected at random from a manufacturing process. Each item is inspected and classified defective (D) or non-defective (N). To list the elements of the sample space, we construct the tree diagram. Sample Space: S {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN} 2.2 Events An event is any collection (subset) of outcomes contained in the sample space S. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome. Example 1: We may be interested in the event A that the outcome when a die is tossed is divisible by 3. This will occur if the outcome is an element of the subset A {3,6} of the sample space S. 3

4 Events Relations from the Set Theory 1-The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement, of A by the symbol A. 2-The intersection of two events A and, denoted by the symbol A I, is the event containing all elements that are common to A and. AI Read: A and A A Events Two events A and are mutually exclusive, or disjoint, if A I φ ie, if A and have no elements in common. 3- The union of the two events A and, denoted by the symbol AU, is the event containing all the elements that belong to A or or both. Read A or AU Mutually Exclusive A A 4

5 Venn Diagrams AU AI A A Mutually Exclusive A A Events Example 1 : Rolling a die. S {1, 2, 3, 4, 5, 6} Let A {1, 2, 3} and {1, 3, 5} A U {1,2,3,5} A I {1,3} {, A {4,5,6} 5

6 Events Example 3: In a Venn diagram we let the sample space be a rectangle and represent events by circles drawn inside the rectangle. EXERCISE 2.1 List the elements of each of the following sample spaces: (a) the set of integers between 1 and 50 divisible by 8. (b) the set S {x x2 + 4x - 5 0}; (c) the set of outcomes when a coin is tossed until a tail or three heads appear. (d) the set S {x a; is a continent}; (e) the set. S {x \ 2x - 4 > 0 and X < 1}. 6

7 Solution 2.1 (a) S {8, 16, 24, 32, 40, 48}. (b) For x 2 + 4x 5 (x + 5)(x 1) 0, the solutions are: x 55 and x 1. So, the sample space S { 5, 1}. (c) S {T,HT,HHT,HHH}. (d) S {N. America, S. America, Europe, Asia, Africa, Australia, Antarctica}. (e) Solving 2x 4 0 gives x 2. Since we must also have x < 1, it follows that S φ Exercise 2 (2.14) Let S {0,1,2,3,4,5,6,7,8,9} and A {0,2,4,6,8}, {1,3,5,7,9}, C {2,3,4,5}, and D {1,6, 7}, List the elements of the sets corresponding to the following events: 7

8 Solution robability of an Event 8

9 2.4 - robability of an Event The probability of an event A corresponds to the occurrence of that event; It is characterized by: If the events A1, A2, A3, are mutually exclusive events, then : Example: A coin is tossed twice (2 times). What is the probability that at least one head occurs? Solution: The sample space; for this experiment is: If the coin is balanced, each of these outcomes would be equally likely to occur. Therefore, we assign a probability of w to each sample point. Then 4w 1, or w 1/4. If A represents the event of at least one1 head occurring, then A {HH, HT, TH} and ( A)

10 Example: let A be the event that an even number turns up and let be the event, that, a number divisible by 3 occurs. Find (A U ) and ( ) A I Solution: For the events A {2,4,6} and {3,6} we have y assigning a probability of 1/9 to each odd number and 2/9 to each even number, we have ( A ) ( A ) robability of an Event Theorem: If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is ( A) n N 10

11 Example: A statistics class for engineers consists of 25 industrial, 10 mechanical, 10 electrical, and 8 civil engineering students. If a person is randomly selected by the instructor to answer a question, find the probability that the student chosen is (a) an industrial engineering major, (b) a civil engineering or an electrical engineering major. Solution: Denote by : M, E, and C the students majoring in industrial, mechanical, electrical, and civil engineering, respectively. The total number of students in the class is 53. all of which arc equally likely to be selected. Since 25 of the 53 students are majoring in industrial engineering, the probability of event: ( I ) Since 18 of the 53 students are civil or electrical engineering majors, it follows that: 18 ( C U E ) Additive Rules Theorem 1: If A and are two events, then ( A U ) ( A ) + ( ) ( A ) Corollary 1: If A and are mutually exclusive, then ( AU ) ( A) + ( ) If A and are mutually exclusive, then ( A I ) 0. 11

12 Additive Rules Corollary 2: If A 1, A 2,..A n are mutually exclusive, then ( A1 U A2... An ) ( A1 ) + ( A2 )... + ( An ) Theorem 2: If A and A are complementary events, then ( A) + ( A') 1 Example: A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability that it is a queen or a heart? Solution: Q Queen and H Heart ( Q ), ( H ), ( Q I H ) Q ( U H ) Q ( ) + H ( ) Q ( I H )

13 Conditional robability For any two events A and with () > 0, the conditional probability bilit of A given that t has occurred is defined d by ( ) A ( ) ( ) A Which can be written: ( ) ( ) ( ) A A Example: Example: Consider the toss of two dice. Let E {sum of spots on dice is 4} F {sum of spots on dice is at most 4}. (E) 1/12 since E {(1, 3), (2, 2), (3, 1)}. (F) 1/6 since F {(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1)}. What about (E F)? E {sum of spots on dice is 4} F {sum of spots on dice is at most 4} ( E F) ( E F) ( F) ( E) ( F)

14 Independence Two events A and are independent events if Or ( A ) ( A). ( / A) ( A) Otherwise A and are dependent. Multiplicative Rule If in an experiment the events A and can both occur, then ( A ) ( A) ( / A) 14

15 Multiplicative Rule Events A and are independent events if and only if ( A ) A ( ) ( ) Note: this generalizes for more than two independent events. Example One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from the second bag is black? 15

16 Thank You Any Questions? STAT 319 robability and Statistics For Engineers Dr Mohamed AICHOUNI & Dr Mustapha OUKENDAKDJI 16

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