In this activity, students analyze their results from the Warm-up to estimate the probability of drawing a name from their bag based on the relative frequency of drawing the name. They also reason that additional trials can more accurately estimate probability.

Some students may have difficulty understanding why the data that they collect from 15 draws from the bag does not match exactly to the contents of the bag. Prompt students to look at the contents of Bag 1 (Clare x3, Mai x5, Jada x7) and to think about a situation in which Clare's name is drawn on the first 3 draws. Emphasize to students that the slip of paper with Clare's name on it is returned to the bag after each draw so it is possible that Clare's name can be drawn a fourth time. If her name is drawn a fourth time, then it is not possible for Mai to be drawn 5 times and Jada to be drawn 7 times because only 11 draws remain.

The purpose of this discussion is for students to use their investigation of a chance process to estimate the probability of an event based on the relative frequency of the event occurring in many trials. Tell the students that the bags have these names inside:

• Bag 1: Clare x3, Mai x5, Jada x7
• Bag 2: Andre x1, Diego x6, Elena x8
• Bag 3: Noah x10, Elena x3, Priya x2
• Bag 4: Clare x10, Mai x3, Jada x2
• Bag 5: Andre x9, Diego x4, Elena x2

Ask the groups to guess which bag they drew from and to explain their reasoning.

Tell students that a convention for writing the probability of an event is to write . For example, would represent the probability of drawing Clare’s name from the bag. When the meaning is clear, it can be shortened to . So, for bag 1, or or 0.2 (or equivalent).

Here are some questions for discussion.

• “Are you sure which bag you have?” (I am pretty sure it is bag 1 because drawing Jada’s name is our highest probability, and her name appears only in bag 1 and bag 4. If we had bag 4, I would have expected Clare's name to be the most frequently drawn name.)
• “Would drawing slips out of the bag 100 times make you more sure of which bag you have?” (Yes, it would make me more confident that I have bag 1, but I was fairly convinced already.)
• “Do you think your list of 10 names from the fourth question matches the list of 10 names created by the other people in your group? Explain your reasoning.” (I do not think it will match because we are not sure which names the teacher will draw, but I would expect that Mai or Clare would be chosen the most frequently.)
• “For the last question of the activity, is it possible for the teacher to draw Jada’s name twice if she draws only ten times?” (Yes, it is possible, but not probable.)

In this activity, students use probabilities of events to suggest possible events that result in the given probabilities. The events in this activity refer to the probability of encountering letters or categories of letters such as vowels. Students think of words for which selecting a random letter from the word results in the given probabilities. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Some students may have difficulty creating their own word to meet the given criteria. For students who need help, consider providing a page from a newspaper or book to help them think of ideas. Prompt students to think about what probability tells them about how many total letters are in the word, how many of the letters are consonants, and how many of the letters are vowels. Emphasize that many words can meet the given criteria. If students continue to struggle, tell them that they may come up with a collection of letters that fit the clues even if it is not a word.

The goal for this discussion is for students to become familiar with using probability to describe chance events. It also provides an opportunity to informally assess how students think about probabilities where “or” or “not” is used.

Ask students, “How do you know your chosen words fit the probabilities given without drawing letters from a hat or using some other chance experiment repeatedly?” (When we know the entire sample space and the number of outcomes in the event, the probability is the number of outcomes in the event divided by the number of outcomes in the sample space.)

Display the word KANGAROOS for all to see. Here are some questions for discussion.

• “If a letter is selected at random, what is the probability that it is a vowel?” ()
• “If a letter is selected at random, what is the probability that it is a consonant?” ()
• “If a letter is selected at random, what is the probability that it is an S?” ()
• “If a letter is selected at random, what is the probability that it is an A?” ()
• “If a letter is selected at random, what is the probability that it is an S or an A?” ()
• “If a letter is selected at random, what is the probability that it is not a K?” ()

If time permits, ask these questions:

• “Whose partner came up with some great clues for question 5?” (Sample response: My partner came up with two great clues. . )
• “What are some words that satisfy the given probabilities?” (Sample responses: just, jump, jack, jars)
• “For the first question, why is FLOWER a valid word?” (It has 6 letters with 2 vowels and 4 consonants. Since and , the probabilities are the same.)