Learn on PengiPengi Math (Grade 7)Chapter 10: Probability Models and Compound Events

Lesson 3: Compound Events: Lists and Tables

In this Grade 7 Pengi Math lesson from Chapter 10, students learn to distinguish between simple events and compound events, then represent sample spaces using organized lists and tables, including area models. Students practice calculating probabilities of compound events by identifying and counting favorable outcomes within these structured representations. The lesson also introduces the concept of independent events as it applies to compound probability.

Section 1

Defining Simple and Compound Events

Property

A simple event is an event with a single outcome or consisting of one experiment. A compound event consists of two or more simple events.

Examples

  • Simple Event: Rolling a single six-sided die and getting a 4.
  • Simple Event: Flipping a coin and getting heads.
  • Compound Event: Rolling a die and flipping a coin.
  • Compound Event: Choosing a marble from a bag, and then choosing a second marble.

Explanation

Understanding the difference between simple and compound events is crucial for determining the correct way to calculate probabilities. A simple event involves just one action, like rolling one die or spinning one spinner. A compound event combines two or more of these simple actions, such as rolling two dice or flipping a coin three times. The methods for finding the total number of outcomes, like the Fundamental Counting Principle, apply specifically to compound events.

Section 2

Representing Sample Spaces with Tables

Property

A two-way table organizes all possible outcomes of two events by listing the outcomes of the first event in rows and the outcomes of the second event in columns. Each cell represents one possible compound outcome, and the total number of outcomes equals the number of rows times the number of columns.

Examples

Section 3

Find Probability Using an Organized List

Property

To find the probability of a compound event using an organized list, first list all possible outcomes in the sample space. Then, count the number of favorable outcomes and the total number of outcomes. The probability is the ratio of these two counts.

P(event)=Number of Favorable OutcomesTotal Number of OutcomesP(\text{event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}

Examples

  • Suppose you toss a coin and roll a six-sided die. The organized list of outcomes is: (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6). The probability of getting tails and an even number is P(Tails and Even)=312=14P(\text{Tails and Even}) = \frac{3}{12} = \frac{1}{4}.
  • A bag has one red (R) and one blue (B) marble. A second bag has one green (G), one yellow (Y), and one orange (O) marble. The organized list for picking one marble from each bag is: (R,G), (R,Y), (R,O), (B,G), (B,Y), (B,O). The probability of picking a blue marble and an orange marble is P(B and O)=16P(\text{B and O}) = \frac{1}{6}.

Explanation

An organized list helps you visualize the entire sample space for a compound event. By systematically writing down every possible combination of outcomes, you can ensure no possibilities are missed. Once the list is complete, you can directly count the total number of outcomes and the specific outcomes that match the event you are interested in. This method provides a clear and straightforward way to calculate the probability by forming a fraction with these counts.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Simple and Compound Events

Property

A simple event is an event with a single outcome or consisting of one experiment. A compound event consists of two or more simple events.

Examples

  • Simple Event: Rolling a single six-sided die and getting a 4.
  • Simple Event: Flipping a coin and getting heads.
  • Compound Event: Rolling a die and flipping a coin.
  • Compound Event: Choosing a marble from a bag, and then choosing a second marble.

Explanation

Understanding the difference between simple and compound events is crucial for determining the correct way to calculate probabilities. A simple event involves just one action, like rolling one die or spinning one spinner. A compound event combines two or more of these simple actions, such as rolling two dice or flipping a coin three times. The methods for finding the total number of outcomes, like the Fundamental Counting Principle, apply specifically to compound events.

Section 2

Representing Sample Spaces with Tables

Property

A two-way table organizes all possible outcomes of two events by listing the outcomes of the first event in rows and the outcomes of the second event in columns. Each cell represents one possible compound outcome, and the total number of outcomes equals the number of rows times the number of columns.

Examples

Section 3

Find Probability Using an Organized List

Property

To find the probability of a compound event using an organized list, first list all possible outcomes in the sample space. Then, count the number of favorable outcomes and the total number of outcomes. The probability is the ratio of these two counts.

P(event)=Number of Favorable OutcomesTotal Number of OutcomesP(\text{event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}

Examples

  • Suppose you toss a coin and roll a six-sided die. The organized list of outcomes is: (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6). The probability of getting tails and an even number is P(Tails and Even)=312=14P(\text{Tails and Even}) = \frac{3}{12} = \frac{1}{4}.
  • A bag has one red (R) and one blue (B) marble. A second bag has one green (G), one yellow (Y), and one orange (O) marble. The organized list for picking one marble from each bag is: (R,G), (R,Y), (R,O), (B,G), (B,Y), (B,O). The probability of picking a blue marble and an orange marble is P(B and O)=16P(\text{B and O}) = \frac{1}{6}.

Explanation

An organized list helps you visualize the entire sample space for a compound event. By systematically writing down every possible combination of outcomes, you can ensure no possibilities are missed. Once the list is complete, you can directly count the total number of outcomes and the specific outcomes that match the event you are interested in. This method provides a clear and straightforward way to calculate the probability by forming a fraction with these counts.