Students must find the probability of a simple compound event. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 35. (author/sw)

Students must compute the probability of a specified event. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 19. (author/sw)

Students are asked to determine the probability of a simple event and explain their answer. This short-answer question is a sample item used in the 2005 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, a complete solution of the problem, and a scoring rubric. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 OGT for Mathematics, Annotated Item 42. (author/sw)

Students are asked to determine the probability that the arrows on two spinners will both land in the same region. Students must justify their answer, and they have the option of using a calculator. This constructed-response question is a sample test item used in grade 12 in the 1996 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1996-12M12, No. 9. (sw)

Students are asked to determine the probability that 2 heads and 1 tail will come up, in any order, if a fair coin is tossed three times. Students have the option to use a calculator. This constructed-response question is a sample test item used in grade 12 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-12M9, No. 17. (sw)

Students are asked to solve a problem involving sampling without replacement.  This constructed-response question is a sample test item used in grade 8 in the 2005 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 2005-8M12, No. 7. (sw)

Students must find the probability of a compound event. This multiple-choice question is a sample item used in the 2005 Ohio Grade 8 Achievement Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 Ohio Grade 8 Achievement Test for Mathematics, Annotated Item 32. (author/sw)

Students are asked to determine the probability of one of Drake’s tickets being selected. This multiple-choice question is a sample item used in the 2008 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OGT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2008 OGT for Mathematics, Annotated Item 43. (author/sw)

Students are asked to find the probability that a randomly selected student will be a boy, given the number of boys and girls in a class. This multiple-choice question is a sample test item used in grade 8 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-8M7, No. 18. (sw)

This lesson presents a classic game-show scenario: You pick one of three doors in hopes of winning a prize. The host opens one of the two remaining doors which reveals no prize, then asks if you want to stick with your original choice or switch to the other unopened door. Which strategy gives you the best chance of winning? Approaches to the solution run from guesses to experiments to computer simulations (links to applets are provided) to theoretical models. The lesson also includes links to appropriate Internet extensions. This lesson plan first appeared in the April 1991 issue of Mathematics Teacher. (author/sw)

This lesson is a step-by-step introduction to tree diagrams for computing probabilities of simple compound events. A Racing Game applet is included, along with discussion questions and suggestions for guided and independent practice. (sw)

Most students understand that the probability of an event occurring can be influenced by another event that has already occurred. However, many students cannot understand that the probability of an event occurring can actually be dependent on an event that occurred later. Having information about the outcome of a later event can be used to revise probabilities of the occurrence of a previous event. This lesson on the time-axis fallacy will help students understand this important, but counterintuitive, idea by developing an understanding of Bayes' Theorem. The lesson plan includes a brief discussion of the mathematical topics, a list of needed materials, a suggested teaching procedure that includes several interesting problem scenarios, and a complete solution guide. (author/pk)

First, students review Pascal's Triangle by completing and discussing the entries of the first eight rows. They then determine the total number of possible girl/boy combinations in a five-child family. This is accomplished by having students investigate the make-up of different five-child families that could be born in a town called Smithville. A coin is used to simulate the births of different children. If the coin shows a head, the child is a girl and if it shows a tail, the child is a boy. The different combinations are presented in an organized manner so that students can discover patterns that will enable them to identify all possibilities. Students are encouraged to look for patterns that will assist them in generating the numbers in subsequent rows of Pascal's Triangle. Finally, students work collaboratively to address and analyze questions regarding the theoretical probabilities of other multiple-child families using Pascal's Triangle. In addition to the lesson plan, the site includes ideas for teacher discussion, extensions of the lesson, and additional resources. The lesson plan is accompanied by video clips illustrating lesson procedures. The user should first locate the Smithville Families lesson and then access the appropriate video clips at the PBS TeacherSource website. The video player necessary to view the video clips can be downloaded for free from the site. (author/sw)

Students investigate the number of people necessary to have a match of the day of the week of birth, month of birth, and date of birth by collecting data from different sizes of groups. A random number generator allows students to simulate this process to estimate the experimental probability of two birthdays matching. Using complementary and independent events, students calculate the theoretical probability of each event. This mathematically rich problem was prepared by the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools developed by the Ohio Department of Education. (author/sw)

This site describes three creative games that give students an opportunity to apply concepts related to probability. Students are asked to determine whether the games are fair, and if unfair, how they might be changed. For many students, exploring situations involving chance can be a challenging and enjoyable mathematical experience. Students can be quite fascinated when their intuitive guesses turn out to be wrong. Complete solutions to the problems are included, and Internet extensions are provided. This resource is adapted from an article that appeared in the April 1992 issue of Mathematics Teacher. (author/sw)