Integrated Algebra Regents Review #2 Geometry Relative Error Probability
Geometry See reference table Formulas you need to know!
Geometry Finding Perimeters In the diagram, ABCD is an isosceles trapezoid. Its bases are AB and CD. BA is extended to E, and DE and EB are perpendicular. Side BC is a diameter of semicircle O, AB = 4, AE = 3, DE = 4, and DC = 10. Find the perimeter of the figure to the nearest tenth. Perimeter = distance around the figure Find the distance around the semicircle a 2 + b 2 = c 2 3 2 + 4 2 = c 2 25 = c 2 5 = c If AD = 5 then BC = 5 Perimeter = 10 + 4 + 3 + 4 + Perimeter = 28.853… = 28.9 units Isosceles Trapezoid AD = BC
Geometry Finding Areas Find the area of the composite figure pictured below. Represent your answer in terms of pi. Area of Triangle + Area of Semicircle A = ½ bh A = ½ (12)(14) A = 84 Remember: An answer left in terms of pi is accurate and exact. Rounding leads to an approximate result. Never round unless otherwise directed.
Geometry Shaded Area Mr. Petri has a rectangular plot of land with a length of 20 feet and a width of 10 feet. He wants to design a flower garden in the shape of a circle with two semicircles at each end of the center circle, as shown in the accompanying diagram. He will fill in the shaded area with wood chips. If one bag of wood chips covers 5 square feet, how many bags must he buy? Area of Rectangle – Area 2 Circles A = lw A = (10)(20) A = 200 Diameter = 10 Radius = 5 Area of Shaded Region 200 – 42.92036…square feet Bags of Wood Chips Needed Mr. Petri will need 9 bags of wood chips to cover the shaded area.
Geometry Surface Area and Volume a.Find the volume of the cylinder to the nearest hundredth. b.Find the surface area to the nearest hundredth of the cylinder if it represents a can which has no lid or bottom. V = lwh (not on the reference table) SA = 2lh + 2hw + 2lw
Relative Error The relative error expresses the "relative size of the error" of the measurement in relation to the measurement itself. Any measurement made with a measuring device is approximate. The error in measurement is a mathematical way to show the uncertainty in the measurement. It is the difference between the result of the measurement and the true value of what you were measuring. Always subtract smaller number from bigger number to create a positive difference.
Relative Error A student mistakenly measures the length of a radius to be 24 inches. The actual radius is 25 inches. a.Find the relative error. b.Find the percent of error. =.04 Percent Error =.04 x 100 = 4% The groundskeeper is replacing the turf on a football field. His measurements of the field are 130 yards by 60 yards. The actual measurements are 120 yards by 54 yards. What is the relative error, to the nearest ten thousandth, in calculating the area of the football field? Actual Area A = lw A = (120)(54) A = 6480 Measured Area A = lw A = (130)(60) A = 7800 The relative error is 0.2037
Probability If there are a ways for one activity to occur, and b ways for a second activity to occur, then there are a b ways for both to occur. Multiply the number of ways each activity can occur. The Counting Principle Examples: 1. Activities: roll a die and flip a coin There are 6 ways to roll a die and two ways to flip a coin. There are 6 2 = 12 ways to roll a die and flip a coin. 2. Activities: a coin is tossed five times There are 2 ways to flip a coin when each coin is flipped. There are 2 2 2 2 2 = 32 arrangements of heads and tails.
Probability The Counting Principle 2) Your state issues license plates consisting of letters and numbers. There are 26 letters and the letters may be repeated. There are 10 digits and the digits may be not be repeated. How many possible license plates can be issued with two letters followed by three numbers? 1)A movie theater sells 4 sizes of popcorn (small, medium, large and extra large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? ways of ordering popcorn Letters: 26 (with repeats) Digits: 0 – 9 (10 total without repeats) license plates
Probability Permutations Consider, four students walking toward their school entrance. How many different ways could they arrange themselves in this side-by-side pattern? 1,2,3,4 2,1,3,4 3,2,1,4 4,2,3,1 1,2,4,3 2,1,4,3 3,2,4,1 4,2,1,3 1,3,2,4 2,3,1,4 3,1,2,4 4,3,2,1 1,3,4,2 2,3,4,1 3,1,4,2 4,3,1,2 1,4,2,3 2,4,1,3 3,4,2,1 4,1,2,3 1,4,3,2 2,4,3,1 3,4,1,2 4,1,3,2 The number of different arrangements is 24 or 4! = 4 3 2 1. There are 24 different arrangements, or permutations, of the four students walking side-by-side. The notation for a permutation: n P r n is the total number of objects r is the number of objects chosen 4 P 4 = 4! Consider the example above: There are 4 friends and all 4 friends are being arranged. A permutation is an arrangement of objects in a specific order. The order of the arrangement is important!!
Probability Permutations Not all permutations are factorials! Find the number of ways to arrange 7 books on a shelf. 7 P 7 = 7! 7 6 5 4 3 2 1 = 5040 ways Find the number of ways to arrange 5 books on a shelf chosen from a set 7 books. 7 P 5 = 7 6 5 4 3= 2520 ways Calculator Corner: To compute factorials (!)… Example: 7! 1)Enter number (7) 2)Press Math 3)Scroll to the right to PRB 4)Press #4 (!) 5)Enter To compute permutations ( n P r )… Example: 7 P 5 1)Enter the 1 st number (7) 2)Press Math 3)Scroll to the right to PRB 4)Press #2 ( n P r ) 5)Enter second number (5) 6)Enter
Probability Theoretical & Experimental Probability Theoretical Probability of an event is the number of ways that the event can occur, divided by the total number of outcomes. Empirical Probability of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences. Ex: What is the probability of landing on an even number if a die is rolled? Sample Space: 1 2 3 4 5 6 Even #’s: 2, 4, 6 Ex: Mary rolled a die 25 times and landed on an even number 9 times. What is the empirical probability that Mary will land on an even number on her next roll?
Probability Theoretical & Experimental Probability Sum of the rolls of two dice 3, 5, 5, 4, 6, 7, 7, 5, 9, 10, 12, 9, 6, 5, 7, 8, 7, 4, 11, 6, 8, 8, 10, 6, 7, 4, 4, 5, 7, 9, 9, 7, 8, 11, 6, 5, 4, 7, 7, 4, 3, 6, 7, 7, 7, 8, 6, 7, 8, 9 Karen and Jason roll two dice 50 times and record their results in the accompanying chart. 1) What is their empirical (experimental) probability of rolling a 7? 2) What is the theoretical probability of rolling a 7? Empirical Probability Theoretical Probability
Probability Sample Spaces Ex: Marnie wants to choose an outfit consisting of a blouse (green or red), a pair of pants (jeans or khakis) and a pair of shoes (sandals or sneakers). Create a sample space to show all the different possible outfits she can make. A sample space is a set of all possible outcomes for an activity or experiment. Green Blouse, Jeans, Sandals Green Blouse, Jeans, Sneakers Green Blouse, Khakis, Sandals Green Blouse, Khakis, Sneakers Red Blouse, Jeans, Sandals Red Blouse, Jeans, Sneakers Red Blouse, Khakis, Sandals Red Blouse, Khakis, Sneakers 8 possible combinations of outfits 1)How many outfits include a pair of jeans? 2)What is the probability that Marnie will choose an outfit with a red blouse or sneakers? 4 outfits GB, Jeans, Sa GB, Jeans, Sn RB, Jeans, Sa RB, Jeans, Sn RB, K, SnGB, J, SnRB, K, Sa RB, J, SnGB,K, Sn RB, J, Sa
Probability Sample Spaces Sample spaces can also be represented using tree diagrams. Ex: Using a tree diagram, create the sample space for tossing a coin 3 times. HHH HHT HTH HTT THH THT TTH TTT a)How many outcomes include two heads and a tail? b)What is the probability of landing on at least two heads out of the three tosses? HHT HTH THT 3 outcomes HHH HHT HTH THT 4/8
Probability Compound Probability If A and B are independent events, then P(A and B) = P(A) P(B). “With Replacement” Example: A drawer contains 3 red paperclips, 4 green paperclips, 5 blue paperclips, 1 white paperclip and 2 yellow paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. a)What is the probability that the first paperclip is red and the second paperclip is blue? b)If the first paperclip is not replaced, what is the probability that first paperclip is red and the second is blue? c)If the first paperclip is not replaced, what is the probability that both paperclips are red? If A and B are dependent events, and A occurs first, then P(A and B) = P(A) P(B, once A has occurred) “Without Replacement” Without Replacement: Denominator decreases!
Probability Conditional Probability The conditional probability of an event B, in relation to event A, is the probability that event B will occur given the knowledge that an event A has already occurred. Example: You toss two pennies. The first penny shows HEADS and the other penny rolls under the table and you cannot see it. What is the probability that they are both HEADS? Sample Space-Tossing two Coins: HHTT HTTH 4 outcomes Based on the information given, the sample space only includes HH and HT.
Probability Conditional Probability GradeSnowboardingSkiingIce SkatingTOTAL 6th684146155 7th845670210 8th597447180 TOTAL211171163545 Middle school students were surveyed about what their favorite sport is. The results are shown in the following table. If a student is selected at random, what is the probability that the student prefers snowboarding given that he/she is in sixth grade grade? Conditional Probability is the same as Conditional Relative Frequency Condition: The student is in 6 th grade What is the probability that the student prefers snowboarding?
Now it’s your turn to review on your own! Use the information presented today to help you practice questions from the Regents Exams in the Green Book. See halgebra.org for the answer keys. Integrated Algebra Regents Review #3 Tomorrow (Tuesday), June 17 th BE THERE!