Probability and Statistics PDF

IT1807 Sample Space, Relationships Among Events and Rules of Probability Probability is used to describe the phenomenon of Examples: chance or randomness of events to occur. It does not deal with guarantees, but with the likelihood of an Experiment Sample Space...

IT1807 Sample Space, Relationships Among Events and Rules of Probability Probability is used to describe the phenomenon of Examples: chance or randomness of events to occur. It does not deal with guarantees, but with the likelihood of an Experiment Sample Space Event occurrence of an event. If we understand how to Select a student The set of The calculate probabilities, we can make thoughtful in your class. students in students in decisions about random and unpredictable situations your class your class where multiple outcomes are possible. Select a student {red, black, red, black, o Probabilities sometimes are subjective (aka in your class brown, blond, brown, theoretical or classical probability) and is based and observe the green,...} blond, on past experience and judgment of the person color of his or green,... to determine whether a specific outcome is likely her hair. to occur. It contain no formal calculations and Choose 2 cars The set of all Collections differ from person to person, and they contain a (without regard collections of 2 of 2 cars high degree of personal bias. There are several to order) at cars chosen chosen from methods for making subjective probability random from a from 10 10 assessments: fleet of 10.  Opinion polls can be used to help in determining subjective probabilities for Simple Event: Probability of getting a Head (H) when a possible election returns and potential coin is tossed; Probability of getting 1 when a die is thrown political candidates.  Experience and judgment relate back to Compound Event: When two coins are tossed, probability upbringing as well as other events the person of getting a Head (H) in the first toss and getting a Tail (T) has witnessed throughout his life. A in the second toss. production manager, for instance, might believe that the probability of manufacturing Sample Problem: a new product without a single defect is 0.85. A manufacturer inspects 50 computer monitors and finds  In the Delphi method, a panel of experts is that 45 have no defects. What is the probability that a assembled to make their predictions of the monitor chose at random has no defects? future. o Other times probabilities are objectively (aka Solution: empirical or experimental probability) based on examining past data and using logical and mathematical equations involving the data to determine the likelihood of an independent event occurring. Probability Formula: Sample Problem: A survey was taken on 30 classes at a school to find the total number of left-handed students in each class. The Where: table below shows the results: P(E) – Experiments: refers a situation involving chance or probability that produces an event. No. of Left-handed students 0 1 2 3 4 5 n(S) – Sample space: refers to set of all possible Frequency (no. of classes) 1 2 5 12 8 2 outcomes of an experiment, that is, any subset of the sample space. A class was selected at random. Find the probability that n(E) – Event: refers to one or more of the the class has 2 left-handed students. possible outcomes of a single trial of an experiment. When one event occurs, it is simple Solution: event. When two or more events occur in a The number of possible outcomes is 30. sequence, it is compound event. 02 Handout 1 *Property of STI  [email protected] Page 1 of 7  IT1807 Basic Properties of Probabilities Property 1: The probability, P, of any event or state of nature occurring lies between greater than or equal to 0 or 0% and less than or equal to 1 or 100%. That is: c. What is the chance of picking the number 1? Solution: Property 2: The probability of an event will not be less Sample space: {1,2,3,4,5,6, 7, 8} than 0 because it is not possible (impossible) or can never occur. That is: Thus, the number of possible outcomes = 8 Event: The number of ways a certain event can occur = 1 (because only number 1 is going to be pick) Property 3: The probability of an event will not be more than 1 because 1 is certain that something will happen (sure event). That is: Sample Problem: Compute for the following items and place them on the probability scale. a. What is the chance of picking a number between 1 - 8? The Addition Rule of Probabilities (Events Involving “OR”) Solution: Sample space: {1, 2, 3, 4, 5, 6, 7, 8} Mutually Exclusive (special addition rule) - The Thus, the number of possible outcomes = 8 probability that A or B will occur is the sum of the Event: One event of this experiment is picking the probability of each event. number 4 for instance between 1 to 8. Not Mutually Exclusive (general addition rule) - The Thus, the number of ways a certain event can occur probability that A or B will occur is the sum of the = 8 (because any number may be pick) probabilities of the two (2) events minus the probability that both will occur. The Multiplication Rule of Probabilities (Events Involving “AND”) o Independent Event (special multiplication rule) - Two events are independent if the occurrence or nonoccurrence of one of the events does not affect the likelihood that the other event will occur. b. What is the chance of picking a number that is even? Sample Problem: Solution: A sock drawer contains one pair of socks with each of Sample space: {1,2,3,4,5,6, 7, 8} the following colors folded together in a matching set: Thus, the number of possible outcomes = 8 blue, brown, red, white, and black. It is an early Event: {2, 4, 6, 8} morning, you are tired and you randomly reach into Thus, the number of ways a certain event can occur the sock drawer and grab a pair of socks without = 4 (because an even number is obtained) looking. The first pair you pull out is the wrong color, which is red. You replace this pair and then choose 02 Handout 1 *Property of STI  [email protected] Page 2 of 7  IT1807 another pair of socks. What is the probability that you will choose the red pair of socks twice? Solution: o Dependent Event (general multiplication rule) - Two events are dependent if the occurrence of one event does affect the likelihood that the other event will occur. Sample Problem: There are 6 black pens and 8 blue pens with a total of 14 pens in a jar. The probability that you will get a black one when you reach in is: 6/14. But what are your chances of getting a black one if you reach in again? Solution: Clearly, the two events are dependent, since taking the first pen affected the outcome of the next attempt. Let A be the event of reaching the first black pen. Thus, Out of 14 total number of pens, 6 are black. Let B be the event of reaching the second pen. Thus, One pen is taken, thus, we are left with 13 pens only. Out of the remaining 13 pens left, 5 are black. 02 Handout 1 *Property of STI  [email protected] Page 3 of 7  IT1807 Counting Rules Useful in Probability Multiplication Principle of Counting objects taken r at a time (where r is a subset of n), an The fundamental principle of counting is often referred to event cannot repeat. as the multiplication rule. The multiplication principle of counting states that: Permutation Formula: o If there are n1 possible number of outcomes/ways for event 𝐸𝐸1 ; and n2 possible number of outcomes/ways for event 𝐸𝐸2,then the possible number of outcomes/ways for both events is (n1 ∗ n2) number of outcomes/ways. Where: This can be generalize to E events, where E is the number n is the number of objects to choose from of events. The total number of outcomes for E events is: r is the number of objects selected o The multiplication principle of counting only works when all choices are independent of each other. If one Sample Problem: choice affects another choice (i.e. depends on There are 12 puppies for sale at the local pet shop. Four another choice), then a simple multiplication is not are brown, four are black, three are spotted, and one is right. white. What is the probability that all the brown puppies will be sold first? Sample Problem: How many lunches are possible consisting of a main Solution: entrée, fruit, and drink if one can select from the Since the order that the puppies are sold is important, this following? problem relates to permutation. Determine n and r a pizza, a sub sandwich, or chicken nuggets as main o n is 12; since there are 12 puppies. entrée; o r is 4; since there are 4 varieties of puppies to choose a banana, apple, orange, or grapes as fruit; and from (brown, black, spotted, white) milk or apple juice as drink. The number of possible outcomes in the sample space is the number of permutations of 12 puppies taken 4 at a Solution: time. To find the number of lunches that can be serve with three main entrées, four fruits and two drinks, the events are described as follows: o 𝐸𝐸1 is for main entrées: Since we have three main entrées (pizza, sub sandwich and chicken nuggets) we can have n(𝐸𝐸1) = 3. o 𝐸𝐸2 is for fruits: Since we have four fruits (banana, apple, orange and grapes) we can have n(𝐸𝐸2) = 4. o 𝐸𝐸3 is for drinks: Since we have two drinks (milk and apple juice) we can have n(𝐸𝐸3) = 2. Each event is independent, thus, by the multiplication There are four brown puppies that can be sold first. Thus, principle, the total number of dinner is: to find the number of ways to get the 4 brown puppies in their specific positions, the events are described as # of Total # of follows: # of # of main different o 𝐸𝐸1 is for first puppy position: Since we have any of the × fruits × drinks = entrée combinati four puppies in the first puppy position we can have 𝐸𝐸2 𝐸𝐸3 s 𝐸𝐸1 ons 𝐸𝐸 n(𝐸𝐸1) = 4. 3 × 4 × 2 = 24 o 𝐸𝐸2 is for second puppy position: Since we have any of the remaining three puppies in the second puppy There are a total of 24 meals available. position we can have n(𝐸𝐸2) = 3. o 𝐸𝐸3 is for third puppy position: Since we have either of Permutation is a counting technique which refers to the remaining two puppies in the third puppy position the arrangement (or ordering) of a set of objects, from we can have n(𝐸𝐸3) = 2. first to last, where the order in which the objects are o 𝐸𝐸4 is for fourth puppy position: Since we have one selected does matter. In a permutation n different remaining puppy in the fourth puppy position we can have n(𝐸𝐸4)= 1. 02 Handout 1 *Property of STI  [email protected] Page 4 of 7  IT1807 Total # # of # of # of of # of first second third fourth points puppy × puppy puppy × puppy = in the position position position position sample 𝐸𝐸1 𝐸𝐸2 𝐸𝐸3 𝐸𝐸4 space 𝐸𝐸 4 × 3 2 × 1 = 24 Therefore, the probability would be 24 divided by the permutation representing the 11,880 possible orderings. There is only one arrangement as follows: Thus, the probability that a permutation of these tiles selected will be in the chosen sequence is 1 divided by the permutation representing the 756 possible orderings. Once an event occurs in a permutation with n objects taken all at a time where some items consist of look-alikes/duplicates and rest are all different, it can occur again. Formula: Where: n is the total number of objects n1 is the first kind nr is the last kind In a permutation with n different objects taken all at a time, n = r. Sample Problem: A box of floor tiles contains the following in random order: Formula: 5 blue (bl) tiles 2 gold (gd) tiles 2 green (gr) tiles in random order. The desired pattern is bl, gd, bl, gr, bl, gd, bl, gr, and bl. If Where: we selected a permutation of these tiles at random, what n (left of P) is the number of objects to arrange is the probability that we would be choose the correct n (right of P) is the number of positions available sequence? for the objects to fill n! is read as “n factorial” Solution: Determine n, n1, n2, n3, n4 Sample Problem: o n = 9; since there is a total of 9 tiles Ramon has five books on the floor, one for each of his o n1 = 5; since blue occurs 5 times classes: o n2 = 2; since gold occurs 2 times Algebra o n3 = 2; since green occurs 2 times Chemistry English Spanish History 02 Handout 1 *Property of STI  [email protected] Page 5 of 7  IT1807 Ramon is going to put the books on a shelf. If he picks the Step 3 books up at random and places them in a row on the same o Find the total number of possible 5-book shelf, what is the probability that his English, Spanish, and arrangements. Algebra will be the leftmost books on the shelf followed by o Determine n(left) and n(right) for 5-book the 2 other books? arrangements:  n(left) = 5; since there are 5 books to arranged Note: English, Spanish, and Algebra books can be  n(right) = 5; since there are 5 positions available arranged in any order provided that they are all in the for placing the books on the shelf leftmost part of the shelf. Solution: Step 1 o Determine how many book arrangements meet the conditions. o Determine n(left) and n(right) for placing the 3 leftmost books:  n(left) = 3; since there are 3 different books (English, Spanish and Algebra) to arranged Step 4  n(right) = 3; since there are 3 positions available o Determine the probability. for the 3 different books to place on the shelf o There are a total of 12 possible combinations for placing the English, Spanish and Algebra books to the leftmost part of the shelf followed before the other 2 books, thus, the probability is: o Determine n(left) and n(right) for placing the other 2 books:  n(left) = 2; since there are 2 other books to Combination is a selection of objects from a arranged collection in any order as oppose to permutations  n(right) = 2; since there are 2 positions available which deal with the ordered arrangements of objects. for the 2 other books to place on the shelf after In a combination in which r objects can be selected the other 3 books from a set of n objects, the selection rules are: o the order of selection does not matter (the same objects selected in different orders are regarded as the same combination); o each object can be selected only once; this implies that you are not allowed any repeat numbers. Combination Formula: Step 2 o Use the multiplication principle of counting to find the number of successes. Where: # of n is the size of the full set Total # r is the number of selected set placement # of placement for of (n-r) is the number of set that was left for the 3 × the other 2 books = succes leftmost 𝐸𝐸2 ses 𝐸𝐸 books 𝐸𝐸1 Sample Problem: 6 × 2 = 12 A high school is planning to put on a “A Chorus Line” musical. There are 20 singers auditioning for the musical. The director is looking for two singers who could sing a 02 Handout 1 *Property of STI  [email protected] Page 6 of 7  IT1807 good duet. What is the probability that Kevin and Phoebe are the two singers who are selected by the director? Solution: o This question involves a combination because the order of the two students selected does not matter. o Determine n and r:  n = 20; since there are 20 singers auditioning for the musical  r = 2; 2 singers will be chosen among 20 singers o The probability of one of the selections (Kevin and Phoebe) would be 1 divided by the combination. References: 12-7 Probability of Compound Events (n. d.) Retrieved from: http://www.ohschools.k12.oh.us/userfiles/223/Classes/197/12.7%20answers.pdf?id=24221 Addition Rule For Probabilities (2018) Retrieved from: https://www.investopedia.com/terms/a/additionruleforprobabilities.asp Addition Rule For Probability (2017) Retrieved from: https://www.mathgoodies.com/lessons/vol6/addition_rules Addition Rules for Probability (2018) Retrieved from: http://slideplayer.com/slide/11601830/ Dependent Events (2018) Retrieved from: https://www.mathgoodies.com/lessons/vol6/dependent_events Empirical Vs Theoretical (n. d.) Retrieved from: https://www.enotes.com/homework-help/what-difference-between-empirical-theoretical-267734 Empirical vs Theoretical Probability (2018) Retrieved from: https://mathbitsnotebook.com/Geometry/Probability/PBTheoEmpirical.html Ferguson Probability (n. d.) Retrieved from: https://quizizz.com/admin/quiz/5756d67a55b60214b89d2ee5 How to Calculate Probability (n. d.) Retrieved from: https://www.wikihow.com/Calculate-Probability Ideas that Lead to Probability (2018) Retrieved from: http://www.shodor.org/interactivate/lessons/IdeasLeadProbability/ Important Formulas - Probability or Chance (2015) Retrieved from: http://www.careerbless.com/aptitude/qa/probability_imp.php Independent Events (n. d.) Retrieved from: https://www.onlinemathlearning.com/independent-events.html Independent and Dependent Events (n. d.) Retrieved from: https://www.northallegheny.org/cms/lib9/PA01001119/Centricity/Domain/771/13.6NG.pdf Mutually Exclusive Events (2018) Retrieved from: http://www.probabilityformula.org/mutually-exclusive-events.html# Probability (n. d.) Retrieved from: http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U08_L4_T1_text_final.html Probability (2018) Retrieved from: http://slideplayer.com/slide/3255149/ Probability Formula (2018) Retrieved from: http://www.probabilityformula.org/# Probability Game for Kids (n. d.) Retrieved from: http://www.kidsmathgamesonline.com/numbers/probability.html Probability Line (2016) Retrieved from: https://www.mathsisfun.com/probability_line.html Probability Notes (n. d.) Retrieved from: https://coursecontent.nic.edu/keolson/math130ko/Concept%20Notes/Probability%20Notes.htm Probability Problems (2018) Retrieved from: https://math.tutorvista.com/statistics/probability-problems.html Probability Questions with Solutions (n. d.) Retrieved from: http://www.analyzemath.com/statistics/probability_questions.html Probability Word Problems (2018) Retrieved from: https://www.onlinemathlearning.com/probability-problems.html Starter (2018) Retrieved from: http://slideplayer.com/slide/7928518/ The Probability in Everyday Life (n. d.) Retrieved from: http://catalogimages.wiley.com/images/db/pdf/0471751413.excerpt.pdf The Real Life Applications of Probability in Mathematics (2017) Retrieved from: http://www.iraj.in/journal/journal_file/journal_pdf/14-358-149822091462-64.pdf Worksheet Practice for Solving Probability Problems (n. d.) Retrieved from: https://www.brighthubeducation.com/homework-math-help/101137-how-to- solve-probability-problems/ 02 Handout 1 *Property of STI  [email protected] Page 7 of 7