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Introduction to Probability

Dice probability

By Michael Avidon, math editor

For Guardians: Performance Expectations (CCSS)

This lesson addresses the following Common Core State Standard (CCSS) for Grade 7:

  • 7.SP.C.5: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Prerequisites

As with any math lessons, a certain amount of basic material and vocabulary (taught prior to grade 7) is assumed. Specifically, it is assumed that the student is familiar with the following material.

  • Reducing fractions
  • Conversion between percentages and fractions

General information

At the lower levels of mathematics, definitions exist for simple, relatively natural things (e.g., triangle, fraction). As one studies mathematics at higher levels, definitions become more technical and more important. Because mathematics needs to be precise, and the subject matter is built upon the definitions, it is important to lay these out carefully.

Probability is a subject that people may have an intuitive grasp of. But before you can make a careful study of it, many basic definitions of abstract things need to be established.

Here is a bird’s-eye view of the material.

The lesson has some fill-in-the-blanks for the student to complete. The correct fill-ins are included here.

STUDENT VERSION: We created a student version to test your child’s understanding | Download a PDF version of this lesson to print out

Lesson

The lesson starts with an imprecise, or plain English, discussion of what probability is. The precise definition of probability is not given until the middle of the second page. Before that precise definition can be given, numerous definitions of more basic concepts are given.

Mathematicians often create vocabulary with words that are used elsewhere, but have very different meanings in mathematics (as opposed to chemistry, say, where very long words unique to that subject are used). One such word is experiment. This bears little relation to an experiment that you might run in a science lab. In a science experiment, you are attempting to determine some unwavering truth (e.g., what is the acceleration due to gravity?). A probability experiment is something with random results, such as picking a card from a well-shuffled deck.

After probability is precisely defined as a fraction, some informal examples are examined on the bottom of the second page to show the following:

  • Probability can be written as a fraction, decimal, or percent.
  • When the probability is ½, an event is neither likely nor unlikely.
  • When the probability is close to 0, an event is unlikely.
  • The greater the probability, the more likely the event.

Page 1 fill-ins: 3; 50

Page 2 fill-ins: 6; 3; 50; 2

Examples 1, 2, and 3 ask the student to find the probability of an event for a simple experiment (tossing two coins or picking a card from a deck).

After these, several things are explained:

  • When the probability is close to 1, an event is likely.
  • In general, probability is a number between 0 and 1, inclusive.
  • When the probability is 0, the event is impossible.
  • When the probability is 1, the event is certain to happen.

Page 3 fill-ins: 4; 4; 52; 1

Exercises 1 through 6 test basic vocabulary. Exercises 7 through 9 prepare students to find probability by merely asking for one step in the process. Exercises 10 through 13 test basic facts. Finally, exercises 14 through 23 are all simple probability problems.

Exercises with answers

Match the descriptions to the vocabulary words.

  1. one possible result – answer: outcome
  2. the set of all possible results – answer: sample space
  3. when you cannot say the next thing that will happen – answer: random
  4. an action that has different random results that you can name – answer: experiment
  5. doing the experiment one time – answer: trial
  6. a description of several possible results or outcomes – answer: event

For each experiment, write the sample space.

  1. Picking a ball out of a box with a red, orange, yellow, and green ball

Answer: {red, orange, yellow, green}

  1. Tossing three coins simultaneously

Answer: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

  1. Rolling two number cubes, each labeled 1 through 6

Answer: {1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-1, 2-2, 2-3, 2-4, 2-5, 2-6, 3-1, 3-2, 3-3, 3-4, 3-5, 3-6,

4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 5-1, 5-2, 5-3, 5-4, 5-5, 5-6, 6-1, 6-2, 6-3, 6-4, 6-5, 6-6}

Fill in the blanks. Each blank could represent a word, a phrase, or a number.

  1. Probability of an event is defined to be the number of ______ over the number of ______. outcomes in the event; outcomes in the sample space
  2. The least value for the probability of an event is ______. Answer: 0

The greatest value is ______. Answer: 1

  1. A probability of 0 means the event is ______. Answer: impossible

A probability near 0 means the event is ______. Answer: unlikely

  1. A probability of 1 means the event is ______. Answer: certain

A probability near 1 means the event is ______. Answer: likely

Find the probability of each event.

  1. Experiment: Pick a card at random from a standard deck of 52 cards.

Event: Picking a diamond   – Answer: 1/4

  1. Experiment: A box has a green, orange, and purple ball and you pick one at random.

Event: Picking a purple ball  – Answer: 1/3

  1. Experiment: Flip two coins.

Event: Getting exactly one head  – Answer: 1/2

  1. Experiment: Roll one number cube.

Event: Getting at least 3  – Answer: 2/3

  1. Experiment: Roll a pair of number cubes.

Event: Getting the same number on the two cubes  – Answer: 1/6

  1. Experiment: Roll a pair of number cubes.

Event: Getting a sum of 9  – Answer: 1/9

  1. Experiment: 100 raffle tickets are sold and you buy all of them. One is picked at random.

Event: You win the raffle.   Answer: 1

  1. Experiment: 100 raffle tickets are sold and you do not buy a ticket. One is picked at random.

Event: You win the raffle.  Answer: 0

  1. You step into an elevator on the first floor of a 10-story building. A random person steps on after you. What is the probability that they are going to the same floor as you?  Answer: 1/9
  2. You step off the elevator in an office building that you have never been to. You are in the middle of a hallway and can go left or right. At both ends of the hall there are halls where you can choose to go left or right. If you choose randomly, and there is only one way to get to your destination, what is the probability you will get there on your first try? Answer: 1/4

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