Slide 1 / 66 Slide 2 / 66 Geometry Probability 2015-10-28 www.njctl.org Slide 3 / 66 Slide 4 / 66 Table of Contents Throughout this unit, the Standards for Mathematical Practice are used. Click on a topic to go to that section MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. Probability of Simple Events MP4: Model with mathematics. MP5: Use appropriate tools strategically. Probability and Length MP6: Attend to precision. MP7: Look for & make use of structure. Probability and Area MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Slide 5 / 66 Slide 6 / 66 Sample Space A sample space is a set of ALL possible outcomes for an activity or experiment. A sample space is usually denoted using set notation {...} and the possible outcomes are listed as elements in the set {a, b, c, ... z}. Probability of Simple Events Return to Table of Contents
Slide 7 / 66 Slide 8 / 66 Sample Space Theoretical Probability Find the sample space in the box below each activity. If the outcomes in a sample space are equally likely to occur, the theoretical probability of an event P (event) is a numerical value from 0 to 1 that measures the likelihood of an event. You can write the probability of an event as a ratio, decimal or a percent. P (event) = number of favorable outcomes {(1,1) (1,2) (1,3) number of possible outcomes (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) H = heads {yellow, · An event with a probability close to 0 is unlikely to occur. Click to (3,1) (3,2) (3,3) T = tails Click to Click to Click to Reveal blue, red, {1, 2, 3, 4, {yellow, Click to (3,4) (3,5) (3,6) · An event with a probability close to 1 is likely to occur. Reveal Reveal green} 5, 6} green, red} Reveal (4,1) (4,2)(4,3) · An event with a probability of 0.5 is just as likely to occur as not. {HH, HT, Reveal (4,4) (4,5)(4,6) TH, TT} (5,1) (5,2)(5,3) Equally Likely to Occur Certain Impossible (5,4) (5,5)(5,6) or not Occur (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} 0 less likely 0.5 more likely 1 Slide 9 / 66 Slide 10 / 66 Non - Geometric Examples There are 7 red marbles and 3 green marbles in a bag. P(card) = 1/ 52 One marble is chosen at random. Write the probability that a green marble is chosen. P (Green) Suppose you choose a card from the deck. What is .... Write as a fraction Write as a decimal Write as a percent P(Heart) = ______ P(3) = ______ P(4 of Spades) = ______ Slide 11 / 66 Slide 12 / 66 Probability Probability 1. Find the sample space for the activity below. 1. Find the sample space for the activity below. 2. Find the probability of each event. 2. Find the probability of each event. P (2) = _______ P (green) = _______ P (even) = _______ P (orange) = _______ P (prime) = _______ P (> 4) = _______
Slide 13 / 66 Slide 14 / 66 Probability 1 A multiple choice question has 14 possible answers, only one of which is correct. Is it "unlikely" to answer a question correctly if a random guess is made? 1. Find the sample space for the activity below. Yes 2. Find the probability of No each event. P (heads) = _______ P (tails) = _______ Slide 15 / 66 Slide 16 / 66 2 What is the sample space for flipping a coin twice? 3 What is the sample space for flipping a coin 3 times? A HT, TH A HHH, TTT, THT, HTH, HHT, TTH, HTH B HH, HT, TH, TT B HHH, HTT, HTH, TTT, HTT, THH, HHT, THT C HH, HT, TT C HTT, THT, HTH, HHH, TTH, TTT D HH, TT, HT, HT D HHH, HHT, HTH, HTT, THH, THT, TTH, TTT Slide 17 / 66 Slide 18 / 66 4 On a multiple choice test, each question has 4 possible 5 A die with 12 sides is rolled. What is the probability of answers. If you make a random guess on the first rolling a number less than 11? question, what is the probability that you are correct? 1/12 A 4 A 10 B 1 B 5/6 C 1/4 C 11/12 D 0 D
Slide 19 / 66 Slide 20 / 66 6 What is the probability of rolling a number greater than 7 What is the probability of randomly choosing a science book from a shelf that holds 3 mystery books, 5 science 2 on a number cube? books and 4 nature books? 1/6 A 1/4 A 1/3 B 1/3 B 1/2 C 5/12 C 2/3 D 7/12 D Slide 21 / 66 Slide 22 / 66 8 A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue? We have evaluated probabilities by counting the number of favorable outcomes and dividing that number by the 1/3 A total number of possible outcomes. 3/16 B 1/13 C In the rest of this unit, you will use a related process in which D 1/7 the division involves geometric measures such as length or area. This process is called geometric probability. click to reveal Slide 23 / 66 Slide 24 / 66 Using Segments to Find Probability Point K on AD is chosen at random. The probability that K is on BC is the ratio of the length of BC to the length of AD . P ( K on BC ) = BC Probability and A B C D AD Length Fill in the blanks. P ( K on AC ) = P ( K on AB ) = Return to Table AD of Contents
Slide 25 / 66 Slide 26 / 66 Using Segments to Find Probability Using Segments to Find Probability Point H on ST is selected at random. What is the probability that H lies Point H on ST is selected at random. What is the probability that H lies on SR ? on SR ? S Q R T S Q R T 2 3 4 5 6 7 8 9 10 11 12 13 14 2 3 4 5 6 7 8 9 10 11 12 13 14 Step 1 : Find the length of each segment. Step 2 : Find the probability. length of SR = ____________ length of ST = ____________ P( H on SR ) = _______________ The probability is _______ or _________% . Slide 27 / 66 Slide 28 / 66 Using Segments to Find Probability Using Segments to Find Probability A point on AM is chosen at random. Find the probability that the A point on AM is chosen at random. Find the probability that the point lies on the given segment. point lies on the given segment. A B C D E F G H I J K L M A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 JL = __________ DJ = __________ Slide 29 / 66 Slide 30 / 66 Using Segments to Find Probability Using Segments to Find Probability A point on AM is chosen at random. Find the probability that the A point on AM is chosen at random. Find the probability that the point lies on the given segment. point lies on the given segment. A B C D E F G H I J K L M A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 BE = __________ AJ = __________
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