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Lesson 11: This AND/OR That

Lesson 11: This AND/OR That

Objective

You will understand how AND/OR probabilities are defined, and you will be able to use frequency tables to compute these probabilities.

Vocabulary

compound probabilities

Essential Concepts

Lesson 11 Essential Concepts

What does "A or B" versus "A and B" mean? These are compound events, and two-way tables can be used to calculate probabilities for them.

Lesson

  1. You have been learning about estimating probabilities of single events based on sample proportions. Today you will learn how to calculate proportions when multiple events happen.

  2. Remember that to compute probabilities, we divide the total number of outcomes we are interested in by the total number of outcomes possible.

  3. Answer the following questions in your IDS Journal:

    1. How would we compute the probability of two outcomes occurring at the same time? For example, in your IDS class, what is the probability that a randomly chosen student likes both pickles AND mayonnaise?

    2. How would we compute the probability of either of two outcomes occurring? For example, what is the probability that a randomly chosen student likes either pickles OR mayonnaise?

  4. Recall that when you calculate the probability of a single event, like a coin toss, the result is a proportion of the number of outcomes of interest divided by the total number of outcomes.

    P(E) = # of outcomes of interest/ # of possible or total outcomes

  5. Next, you will learn how to calculate the probability of multiple events, which we call AND/OR. To do this, you will survey at least 10 people (you may survey more) and keep track of their food preferences.

  6. In your IDS Journal, make a table with 4 columns as shown in the picture below.

  7. You will use this table to keep track of the food preferences of the people you survey. Next, you will ask each person whether that person like pickles, mayonnaise, both, or none. You'll then record each person's answer in the appropriate column that represents their preference. For example, if Joe chooses pickles, then write Joe's name in the Pickles column. If Maria likes pickles and mayonnaise, then write Maria's name under both. The same applies to the other two choices.

    Note: If after you've finished surveying everyone you don't have at least one name in each column, write your name in the column that is blank. Now you will have at least one person in each column.

  8. Under your completed table, record the total number of people you surveyed. In other words, how many names are recorded in your table?

  9. Refer to your table for the items below and record any calculations or numbers in your IDS Journal.

    1. To find out how many people like BOTH pickles and mayonnaise, locate the BOTH column and count the number of people in it. Write down this number.

    2. To calculate the probability that a randomly selected person from your survey likes BOTH pickles and mayonnaise, divide the number of people who like BOTH by the total number of people you surveyed.

      P(BOTH) = number of people under "Both"/ total number of people

  10. Again, refer to your table for the items below and record any responses in your IDS Journal.

    1. How many people from your survey like pickles? What do you think is the answer?

    2. If the count for your answer in (a.) only included the number of people in the Pickles column, that is INCORRECT because people who like BOTH also belong to the group of people who like pickles.

    3. How would you calculate the probability that a randomly selected person from your survey likes pickles?

    4. If you concluded that the probability would be the number of people under Pickles plus the number of people under Both divided by the total number of people surveyed, you are correct.

      P(pickles) = (number of people under "Pickles") + (number of people under "Both") / total number of people surveyed

  11. Refer to your table for the items below one last time and record any responses in your IDS Journal.

    1. How many people from your survey like pickles OR mayonnaise? What do you think is the answer?

    2. If the count for your answer in (a.) only included the number of people in the Pickles and the Mayonnaise column, that is INCORRECT because people who like BOTH also belong to the group of people that like pickles OR mayonnaise.

    3. How would you calculate the probability that a randomly selected person from your survey likes pickles OR mayonnaise?

    4. If you concluded that the probability would be calculated as the number of people under Pickles plus the number of people under Mayonnaise plus the number of people under Both divided by the total number of people surveyed, you are correct.

      P(pickles) = (number of people under "Pickles") + (number of people under "Mayonnaise") + (number of people under "Both") / total number of people surveyed

  12. AND/OR probabilities are called compound probabilities. In your IDS Journal, record your own definition of AND/OR probabilities based on the activity you just completed.

    Sample definition: A compound probability is the probability of some combination of events occurring.

  13. Now you will create a 2-way frequency table like the one below to organize the number of people according to their preferences. Make sure you include the totals in the margins (edges) horizontally and vertically. You should complete the table as follows:

    • Number of people who like both go in the Yes/Like Mayonnaise and Yes/Like Pickles box.

    • Number of people who like none go in the No/Like Mayonnaise and No/Like Pickles box.

    • Number of people who like only mayonnaise go in the Yes/Like Mayonnaise and No/Like Pickles box.

    • Number of people who like only pickles go in the No/Like Mayonnaise and Yes/Like Pickles box.

  14. Instructions: In the handout below, you will complete a table that will contain information about the ice cream preferences of students in an IDS class. To complete it, follow the instructions in items # 15-19. Then answer the questions in the handout by refering back to the table.

    Click on the document name to download a fillable copy of the Compound Probabilities handout (LMR_2.13).

  15. The students in an IDS class received sticky notes to write down which ice cream flavor each preferred. Females received a blue sticky note and males received a gold sticky note.

  16. On their sticky note, each student indicated their ice cream flavor preference: vanilla, chocolate, or rocky road.

  17. When the teacher asked females to hold up their sticky notes, she counted 24. On the table in handout, find the box under the TOTAL column to record the number of females.

  18. When the teacher asked males to hold up their sticky notes, she counted 18. On the table in handout, find the box under the TOTAL column to record the number of males.

  19. Complete the rest of the table in the handout using the information below:

    1. The number of females that like rocky road is 2 and the number that like vanilla is 8.

    2. The number of males that like chocolate is 4 and the number that like vanilla is 5.

    3. Make sure the numbers across (rows) for each flavor match the totals for each gender.

  20. Wow that the table is complete, answer the questions on the Compound Probabilities handout (LMR_2.13). Remember to use what you learned from the food preferences activity to help you calculate compound probabilities.

Reflection

What are the essential learnings you are taking away from this lesson?

Homework & Next Day

Finish the Compound Probabilities handout.

Complete LAB 2D: Queue It Up! prior to the Practicum.