Intent

Students examine a variation on the one-and-one situation in which the probabilities change in the second stage. This continues their preparation for the analysis of the game of Little Pig.

Mathematics

This activity involves conditional probability, in which the probability of the outcome at one stage depends on the outcome of a previous stage. The mathematical goal is to deepen students’ intuition about and analytic ability to work with probability rather than to derive formal algorithms or computation methods. [link to math maps]

Progression

After individually exploring the multistage probabilistic situation posed, students will share approaches. The activity concludes with the naming of this type of problem as a question of conditional probability. Students will likely return to this topic in a few days when they consider Martian Basketball.

Approximate Time

20 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

Prior to assigning this activity, you might help students recall that their initial approach to analyzing the situation in The Theory of One-and-One was to consider a large number of trials and draw an area model.

Discussing and Debriefing the Activity

Ask for volunteers to present their results. Elicit at least one presentation involving area models. Area models provide a visual representation of what is happening and can help many students reason through multistage probability problems.

Students might use a sequence of diagrams to describe the stages of the situation. The following diagrams are based on a consideration of 100 one-and-one situations. This diagram represents the first shot.

Figure 1

This diagram represents the first and second shot.

Figure 2

Thus, Shelly scores no points 20 times, one point 8 times, and two points 72 times. Confirm that 20 + 8 + 72 = 100.

In answer to Question 1, the diagram shows that she scores no points 20% of the time, one point 8% of the time, and 2 points 72% of the time.

For Question 2, students can use the diagram to figure total points. For 100 cases, Shelly scores a total of

20 · 0 + 8 · 1 + 7 · 2 = 152 Points

yielding an expected value of 1.52 points per one-and-one situation.

Explain that this situation is an example of conditional probability, since the probability of Shelly making a free throw depends on when it comes in the shooting sequence.