If students struggle to interpret the new notation , consider providing additional examples to reinforce the idea. Remind them that the event following the vertical line represents the condition we know to be true. It can be considered as the smaller group of outcomes within the sample space being considered for this probability.

For example, when rolling a number cube, means that we consider only the 3 even numbers on the number cube as possible outcomes, and then we determine the probability that it is a 6. With this understanding we can figure out that .

The purpose of this discussion is to understand how to calculate conditional probability and how conditional probability is related to independence. Ask students, “Does Kiran's equation always work? Explain your reasoning” (I think it always works because the left and right side of the equation are really two different ways of saying the same thing. For both events A and B to happen, you can think of first event B happening, then have event A happen under the condition that B has already happened. Multiplying the probability of event B by the probability for A given B, you get the same as the probability of events A and B both happening.). Confirm with students that Kiran’s equation does always work.

Ask:

• “A box contains 8 crayons: 3 red, 2 blue, 1 black, 1 green, and 1 yellow. Remove one crayon from the box at random, and set it apart. Then remove a second crayon from the box. Use Kiran’s equation to find . Explain or show your reasoning.” ( because and .)
• “How is the probability of choosing the second crayon dependent on the first crayon?” (It is dependent because the probability that the second crayon is blue could be or depending on whether or not the first crayon is blue.)
• “How could you represent the crayon situation using a tree diagram?” (When you list out the probabilities of the first crayon, only 2 of the 8 are blue. For each of the picks there are 7 choices so I know there are 56 total possibilities. For each of the 2 picks that were blue, 1 out 7 of the crayons that remain are blue, and 2 out 56 is equal to .)

Some students may not recall what it means for events to be independent or dependent. Ask students to look in the lesson summaries for the unit to find the definition. Then ask the students to give examples of pairs of events from the same experiment that they have seen during the unit and to describe whether they are independent or dependent.

The purpose of this discussion is for students to understand the relationship of to for independent events.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “Are any of these probabilities the same? Explain your reasoning.” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”
• “How do you know ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, ask these questions for discussion.

Here are some questions for discussion.

• “Think of two independent events A and B. What is and ?” (I thought about event A as pulling a name out of a bag with ten different names in it and event B as drawing a face card. is and is also .)
• “If A and B are independent events, what is the relationship of to ? Explain your reasoning.” (They are equal because the probability of A happening is not influenced by whether B occurs or not.)
• “When A and B are dependent events, what is the relationship of to ? Explain your reasoning” (When the events are dependent, the probabilities are not equal because event B affects the probability of event A. If event B did not affect the probability of event A, then the events would be independent.)
• Look at the values for and that you found in questions 2 and 3. Why are the values equal? (These are equal because the probability of flipping heads is the same regardless of what happened with the number cube.)
• Look at the values for and that you found in questions 2 and 3. Why are the values equal? (These are equal because the probability of getting a 4 is not changed by the coin showing tails.)

Display the statement for all to see: “If events A and B are independent, then and .”