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High School Probability

Learning Module

Standards and Objectives

This Learning Module is meant to help high school students learn the mathematical/ statistical concept of probability. This concept is covered within the Common Core class known as “Probability Basics”, which is taken by grade 9 to grade 12 students in high school. The updates in this module will include videos and screenshots from various institutions, which is a great way to learn mathematics. If you are interested in learning more than just how to understand and apply basic concepts of probability, then this is a great place to start.

 

Expectations

By the end of this Learning Module, students in grade 9-12 will be able to:

  • Understand the concepts of basic probability using various examples. 
  • Understand the concept of experimental probability.
  • Understand the difference between theoretical and experimental probability.
  • Understand how to compute the probability using the different methods.
  • Learn the concept of teamwork as they work through their final occupations project

Probability (which uses existing data to predict future events) is one of the branches of math needed for Common Core Standards that high school students can see at work in the world.

Introduction to Theoretical Probability

Media embedded July 18, 2018

 

1. Probability Basics

For Student

Probability is simply how likely something is to happen. Whenever we are unsure about the outcome of an event, we can talk about the probabilities of certain outcomes: How likely they are. The analysis of events governed by probability is called statistics. The best example for understanding probability is flipping a coin. There are two possible outcomes: heads or tails.
What is the probability of the coin landing on Heads? We can find out using the equation P(H)=?. You might intuitively know that the likelihood is half/half, or 50%. But how do we work that out?
 

Probability =

In this case:


Probability of an event = (# of ways it can happen) / (total number of outcomes)


P(A) = (# of ways A can happen) / (Total number of outcomes)


Example 1
There are six different outcomes of rolling a die.


What is the probability formula of rolling a 'one' on a die?

What is the probability of a one or a six outcome when rolling a die?

Using the formula from above:


What is the probability formula and solution of rolling an even number on a die (i.e., rolling a two, four or a six)?


Tips:

  • The probability of an event can only be between 0 and 1 and can also be written as a percentage.
  • The probability of event A is often written as P(A).
  • If P(A) > P(B), then event A has a higher chance of occurring than event B.
  • If P(A) = P(B), then events A and B are equally likely to occur.

For Teacher

Purpose: If the high school students do not understand the probability basics, they will easily fall behind at some point within this learning module. This portion of it, is meant to help the students refresh their mind on what probability is and looks like, and also act as a “warm up session.”

Method: The students will go through the probability basics step by step . You will then be able to gauge their level of prior knowledge and current understanding of the concept at hand. This will enable you to know the pace at which the next learning steps in the module should be carried out.

Tips: You must carefully watch the students to see their reactions as you go through each step and welcome any questions from them. There may be those that may seem to be struggling, as well as those who help you go through this probability basics showing their level of understanding. Ensure the students have a pen and notebook so as for them to list the formulars down for revision purposes. 

2. Simple Probability: Yellow Marble

For Student

Media embedded July 18, 2018

This is just an additional example to ensure that you have understood the basics required so as to understand the concept of probability. Now in this case, it will be marbles and not coins or dies. This should enable you to grasp the introductory stages of the lesson and put you in a better place to understand the latter. You are also encouraged to ask as many questions as possible so as to become better at the concept.

For Teacher

Purpose: If the high school students have understood the probability basics in the previous session, they will easily understand you at this point within this learning module. This portion of it, is meant to help the students practice a little more so as to prepare for a practice test.

Method: The students will go through the slides step by step with your guidance . You will then be able to gauge their level of current understanding of the concept at hand. This will enable you to still know the pace at which the next learning steps in the module should be carried out.

Tips: You must carefully watch the students to see their reactions as you go through each step and welcome any questions from them. There may be those that may seem to be struggling, as well as those who help you go through this probability basics showing their level of understanding. Ensure the students are still listing down the formulars down for revision purposes. Encourage them to ask questions so as to make this lesson be a success in the end.

3. Simple Probability Practice Test

For Student


1) You spin the spinner shown below once. The spinner has 4 equal sectors colored: Blue, red, green and yellow.

What is P(green)?
If necessary, round your answer to 2 decimal places.

 

 2) The probability that player A will win the tennis match is ⅛. What is the probability that player A will lose?

 

3) A letter is chosen at random from the word: MATHEMATICS

What is the probability that :

  • It is an A?
  • It is not a T?
  • It is a vowel?

4) You roll a dice.What is the probability of getting a number less than 10?

 

5) A box contains balls. 3 black, 4 white and 5 red. A ball is drawn at random. Find the probability of:

  • Drawing a black ball?
  • Drawing a yellow ball? 
  • Not drawing a black ball?

For Teacher

Purpose: To see how well the students have grasped the concept of probability so far so as to gauge their level of understanding

Method: The students will work on the five problems. You will then collect the problem and grade them, not to be returned to the students. The problems should be gone over with the class as a whole after it is collected and graded. Grading this, should not have any sort of point value attached to it, but instead be a gauge of how much of the concept has been understood so far. The teacher should then pair up the students in groups of 3. These groups will be used at certain points throughout the module. Each pairing will contain students who have a great understanding of probability basics already, and those who do not. This way, sessions and tests in this course should run smoothly as they will be group-based works. Try to make the groupings as even as possible so no one group is too strong or too weak quantitatively. The grade on this basic knowledge test can be a help in creating these pairings.

Tips: You must navigate the classroom, watching for students that seem to be struggling, as well as those you finish early. This will give you a quick idea of who may struggle and who may not, even if all the grades turn out fairly well. Just because a student gets a decent grade does not mean that they cannot struggle. They may not realize how to solve a basic probability problem immediately, resulting in a potential future problem.

4. Experimental Probability

For Student

Media embedded July 18, 2018

This is just to introduce a new aspect of probability: Experimental Probability. When asked about the probability of a coin landing on heads, you would probably answer that the chance is ½ or 50%. Imagine that you toss that same coin 20 times. How many times would you expect it to land on heads? You might say, 50% of the time, or half of the 20 times. So you would expect it to land on heads 10 times. This is the theoretical probability.

The theoretical probability is what you expect to happen, but it is not always what actually happens. The table below shows the results after Sunnie tossed the coin 20 times.

The experimental probability of landing on heads is: 

It actually landed on heads more times than we expected. Now, Sunnie continues to toss the same coin for 50 total tosses. The results are shown below.

Now the experimental probability of landing on heads is:



The probability is still slightly higher than expected, but as more trials were conducted, the experimental probability became closer to the theoretical probability.

Let us Review:

Theoretical probability is what we expect to happen, where experimental probability is what actually happens when we try it out. The probability is still calculated the same way, using the number of possible ways an outcome can occur divided by the total number of outcomes. As more trials are conducted, the experimental probability generally gets closer to the theoretical probability

For Teacher

Purpose: If the high school students have understood simple probability in the previous session, they will easily understand experimental probability at this point within this learning module. This portion of it, is meant to help the students differentiate between theoretical probability and experimental probability.

Method: The students will go through the slides and the explanation with examples step by step with your guidance . You will then be able to see if their level of current understanding is at per with your expectations. 

Tips: You must carefully watch the students to see their reactions as you go through each step and welcome any questions that they may have. There may be those that may seem to be struggling, as well as those who help you go through this probability basics showing their level of understanding. Ensure the students are still listing down the formulars down for revision purposes. Encourage them to ask questions so as to make this lesson be a success in the end.

5. Experimental Probability Practice Test

For Student

1) Find the experimental probability:
Roll dice: 1, 3, 3, 4, 4
P(1) =

a) 60%
b) 16.7%
c) 40%
d) 20%

2) Find the experimental probability:
Roll dice: 1, 3, 3, 4, 4
P(3) =

a) 60%
b) 17%
c) 40%
d) 80%

3) Experimental Probability is:

a) What will happen
b) What actually happens
c) What should happen
d) What I think happens

4) If you spin the spinner 90 times, how many times should the number 3 be selected? (hint: set up a proportion.)

a) 9
b) 15
c) 16
d) 30

5) Based on the chart, what is the experimental probability of pulling a green marble?

a) 1/8
b) 8/25
c) 8/100
d) 7/8

For Teacher

Purpose: To allow students to practice how to solve experimental probabilities as groups. As all teachers already know, there is most likely more than one way to solve a math problem, in this case there is the 'probability tree' method and the method already shown in the module. This also gives students a choice for which method they prefer the most. Not every learner is created equal, so a variation in solution methods is ideal.

Method: Based on the video you have watched with your class, discuss any questions that may have arisen. Then, once the test is completed, collect it. As before, setting a point value for this test is up to you. Partial credit may be awarded if desired to motivate the students.

Tips: Promote students to be group learners and independent academics. Each person learns in a different way, and express this to your students. Show them that no one method is correct.

6. Theoretical and Experimental Probabilities

For Student

Media embedded July 18, 2018

The probability of an event is a number from 0 to 1 that measures the chance that an event will occur. In this lesson, we will look into experimental probability and theoretical probability. The following table highlights the difference between Experimental Probability and Theoretical Probability.

  • How to find the Experimental Probability of an event?

Step 1: Conduct an experiment and record the number of times the event occurs and the number of times the activity is performed.
Step 2: Divide the two numbers to obtain the Experimental Probability

  • How to find the Theoretical Probability of an event?

The Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.

  • What is the Theoretical Probability formula?

The formula for theoretical probability of an event is:

 

For Teacher

Purpose: To give students a well elaborated explanation on the differences between experimental and theoretical probability. Also, if a student really wants to solve the problem in the last module, this will help them get away from the “guess and check” method, promoting the idea that a system can be solved efficiently; this may be thought to be useful because of the video and sufficient explanation provided.

Method: Have the students get in the groups you have set up to work on the tasks. Let them go through this one more time after you have given a lecture on it. Allow them to ask questions because there might be some who may be a bit confused at this point. Let them digest the module this far, as the end is near.

Tips: Please take the students reactions and previous tests done seriously. Also, promote your students to be honest about the difficulty of this method. Even though the pairings are meant to get the tests finished quickly, some students may just “tag along”, watch the quantitative students do the work, and not gain an understanding of what is going on.

7. Probability Tree Diagram Method

For Student

Due to some students thinking that the other method may be difficult. They should not lose hope at all. This is because there is another method that they may find to be a bit easier, the 'Probability Tree Diagram Method'.

  • What is a Probability Tree Diagram?

We can construct a probability tree diagram to help us solve some probability problems. A probability tree diagram shows all the possible events. The first event is represented by a dot. From the dot, branches are drawn to represent all possible outcomes of the event. The probability of each outcome is written on its branch.

Example:

A bag contains 3 black balls and 5 white balls. Paul picks a ball at random from the bag and replaces it back in the bag. He mixes the balls in the bag and then picks another ball at random from the bag.

a) Construct a probability tree of the problem.
b) Calculate the probability that Paul picks:

  •  two black balls
  •  a black ball in his second draw

a) Check that the probabilities in the last column add up to 1.

b) i) To find the probability of getting two black balls, first locate the B branch and then follow the second B branch. Since these are independent events we can multiply the probability of each branch.

ii) There are two outcomes where the second ball can be black.

Either (B, B) or (W, B)

For Teacher

Purpose: To allow students to practice how to solve experimental probabilities as groups and using a different method, just incase the other method seems to bring difficulties. As all teachers already know, there is most likely more than one way to solve a math problem, in this case there is the 'probability tree diagram' method and the method already shown in the module. This also gives students a choice for which method they prefer the most. Not every learner is created equal, so a variation in solution methods is ideal.

Method: Based on the example you have given, discuss any questions that may have arisen. 

Tips: Promote students to be group learners and independent academics. Each person learns in a different way, and express this to your students. Show them that no one method is correct.

8. Peer Reviewed Project

For Student

“How can probability be used in the various occupations?” 

Almost every student has said this at one point. Take a look at the links provided to see jobs that use mathematics and statistics, and probability specifically:

Project Description: With your groups, pick an occupation from the links and create a slideshow presentation to present to the class. Your group must go out and interview atleast one or two people of the chosen occupation and question them on the mathematical and statistical (probability to be precise) components of their job.

The guidelines for the slide elements are as follows:

  • You must include why you chose to research this occupation.
  • Information about the occupation, such as the expected amount of jobs created in the next couple of years (the US Bureau of Labor is a great place to start to gather information).
  • Provide ways probability concepts are used every day in this occupation. Have the interviewee give a scenario that they recently worked on involving probability.
  • It must contain a slide giving personal information of the person interviewed.
  • Students must also be willing to take questions from the rest of the class on their project at the end.

For each of the first 4 guidelines above, so you should have a slide covering each, your classmates will be grading you according to the following rubric:

1 – “The slide does not look clean, has too many words, and is not related to the content necessary. No media contained either.”

2 – “There are 1-2 quick points that are relevant, but no media for support. Too many words on the slide.”

3 – “There are 3-4 quick points, as well as media. The wording is very precise, and the group has done most of the talking to describe their project. They still sometimes read word-for-word what is on each slide.”

4 – “There are at least 5 points, and media such as links and images to make the slide stand out. The students has great knowledge on their project, and show this by expanding on the facts and details in each slide. Very little, if none at all, reading word-for-word from each slide.”

When you are grading, please write the score for each slide, and then total them up, out of 16.

The average of your classmates’ scores will account for 50% of your project grade, and the other half will come from the teacher-designed rubric.

For Teacher

Purpose: To have students see what they are learning is very applicable to everyday life. The lack of application of math or statistics content in the classroom hurts the field, as students do not get an understanding of why it is necessary for their future lives.

Method: The project should take about four weeks, with the first three weeks for taking the time to gather the data and conduct the interview, and the last week for putting it all together. The point value of the project should be worth the total of the point values from the previous tests as it should be at least half of the class grade.

Tips: The interview meeting should be facilitated to make sure that it happens in a timely manner, as it may be hard for both parties to find a time to conduct the interview. You may have meetings halfway through the project with students to track their progress. Some meetings might even have to be more often to make sure students are going to finish on time. The meetings are also meant to address any student concerns, such as certain group members not contributing equally to the project.

Below is the rubric you will use to grade the student projects. The student rubric is listed as points because they are just meant to total up the scores. It also is meant to only cover the objective of making the slides media-heavy, and not word-heavy. Your rubric is more based on the fluidity of the presentation.

 

9. Module Survey

For Student

Please complete the survey attached to the learning module. Feedback will be highly appreciated.

 

For Teacher

The survey attached is also for you, the educator, to complete on what you thought about this learning module.