Compound Probability

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Objective

SWBAT identify the sample space of compound events by listing samples in a table

Big Idea

students work in groups and pairs to explore compound probability, focusing on identifying sample space

Do Now

10 minutes

Students enter silently according to the Daily Entrance Routine. They are handed their Do Now assignment at the door. A timer is displayed at the front of the room to show that they have 5 minutes to complete the 3 questions on their paper.

The creation of this Do Now is based on student feedback on the topics they are struggling with the most. Two common questions keep coming from students:

  • How do you know when to answer with a fraction or a whole number?
  • What is the difference between experimental and theoretical probability?

The first two items attempt to address these two questions. In question #1 we have an answer in the form of a whole number and for question #2 it is in the form of a fraction. When reviewing the answers, ask students:

  • What is the question asking you to find?
  • Is it asking you to find the probability or the number of times an outcome will occur?
    • If it is probability, the answer must be in the form of a fraction. Why?
    • What is different about questions #1 and #2? What are you asked to find in one vs. the other?
    • For #2, how could you change the question so that it asks about theoretical probability instead of experimental?

My intention when including the third question was to set up a conversation about sample space with students. I initially thought I should give them the freedom to organize the information any way they wanted. However, my experience with this particular group of students this year has been that when there is less structure and more freedom in ways of showing work, many feel overwhelmed and simply do not complete the problem, especially after completing two other problems. By giving them the table they have a structure to use for organizing their information. Some students will inevitably have questions on what they are supposed to do, what is meant by pairs, or a combination of two coins.

Once we review, I make sure to have enough time (at least 3 minutes) to discuss this problem. The big question is:

How many times do we count combinations like penny/nickel? Is this one outcome? Or is there more than one outcome named penny/nickel because there are 5 pennies and 4 nickels?

This is a complex idea that impacts the sample space in a compound probability item. It is likely that I will need to solely plant the seed of this essential question and continue addressing the answer in the class notes (next section).

Class Notes

15 minutes

Students are asked to take out a blank sheet of paper. On it I will have them draw two pictures: a spinner with 4 equal pieces named A, B, C, and D AND the net of a six sided die numbered 1 – 6. The use of manipulatives could be useful here depending on the resources available.

 

 

What if we spin the spinner once and then roll the die once?

 

I ask students to take 5 minutes to create another table like the one in the Do Now to show all the possible outcomes. They may work in pairs. At the end of these 5 minutes, I ask the pairs to find another pair and compare answers. Each group must strive to agree on the number of possible outcomes and review what all the possible outcomes are. The Mathematical Practice in action here is MP7, looking for and making use of structure. By using their table, students are identifying the sample space for this compound event before they know the terms (compound probability, samples space).

 

While students were reviewing answers in groups of 4, I asked one pair to write their answers on the board. This pair should have events listed in a comprehensive way (A1, A2, A3… etc) and all the correct combinations. Once we have the answers, I ask all students to return to their seats. I ask the question:

 

How is this problem different and the same as the Do Now example 3?

 

I’m looking for the two main ideas:

  • They both involve combination of two events (two coins are selected, a spinner is spun and a die is rolled)
  • The problem in the do now has multiple outcomes that appear to be the same (multiple pennies, multiple nickels etc)

 

Bring back the essential question during the Do Now:

 

How many times do we count combinations like penny/nickel? Is this one outcome? Or is there more than one outcome named penny/nickel because there are 5 pennies and 4 nickels?

It is important to push students to understand that getting Penny1 and Nickel1 is a different event than Penny2/Nickel1, despite the fact that it is a penny and a nickel both times. This idea can be illustrated by continuing to make comparisons between that question and the notes sample:

  • Should the number of outcomes change if I change the way I ask the question? No, the number of outcomes only changes when we add things, like pennies or parts of the spinner.
  • What if I asked, “how many different outcomes of a letter and a number on the spinner are there”? or, what if I renamed the parts of the spinner A, A, A, A and the die 1, 1, 1, 1, 1, 1. Does this change the number of outcomes? No, there are still 4 and 6

 

The idea of sample space, and distinct events, is a complex one for students. Given time constraints this year, I am not spending as much time as I would like to with this concept. Ideally I would give this concept alone 3-4 days to itself, having students conduct different experiements with different types of compound events (i.e. coins, spinners, dice with more than 6 sides, clothing and food combos, etc) AND different ways of organizing the information (i.e. tables, tree diagrams, lists, etc).

 

In this lesson we only had time for the one experiment exposing students to listing sample spaces. In the last 5 minutes of this section I ask students to report probabilities, reminding them that the answers are in the form of a fraction where the denominator is the number of total outcomes in the sample space, and the top value is the number of favorable events. 

Class Work

20 minutes

Students receive their Classwork, which is heavy on reading in this lesson. The first 10 minutes are to be spent reading and highlighting important terms in each paragraph included in the sheet with neighbors. Students must be motivated to ask and answer questions as I walk around, ensuring they understand the readings.

  • What is simple probability?
  • How many events are involved in simple probability?
  • What is a “favorable outcome”?
  • What is compound probability? Why is it different than simple probability?
  • How many events are involved in compound probability?
  • What is different about this table [in the CW] and the tables we’ve created in the class notes and do now?
    • How does this table help you organize the information in a better way than the tables we created in the notes and do now?
    • What is a sample space?
    • What does consecutively mean?
    • Is it possible to consecutively draw two marbles of the same color? Why? Yes, because the marbles are being replaced
    • Is red, green the same as green, red? How does order play a role in this problem? No, these are two different events. The question asks about drawing red first and then green, this is only possible once in the table.

Replacement is such a pivotal idea here, warn student to look out for this piece of info in each word problem. I don’t dive deeply into why this replacement concept matters in an intro lesson. This is something I would spend more time exploring if we had more instructional time in the year.

Closing

10 minutes

In the last ten minutes of class we review the answers and focus on the last questions of the classwork: the last “indie try” and the question before it:

We can also determine compound probability using an arithmetic strategy.

Ask:

  • What is “an arithmetic strategy”? using math; avoiding the tables and figuring out a formula for it (MP7)
  • After using the table, what is the probability of drawing a red and then a green marble? Why? Explain each fact in your ratio P(red, green) = 1  / 16
    • Now look at the numbers closely: what is the probability of drawing one red marble (simple probability) 1 / 4
    • What is the probability of drawing one green marble (simple probability)  1 / 4

I push students to see the connection; more specifically discover the rule or algorithm for calculating compound probability (multiply the simple probabilities of each favored outcome).

After answering questions, I distribute Homework and dismiss students.