Intent

This activity presents another simpler game to help prepare students to analyze Big Pig.

Mathematics

This variation of Pig uses coin flips. In Pig Tails, if you get tails, your turn is over. Students will evaluate strategies that include flipping only once each turn, flipping twice each turn, and so on, and then look for a pattern to determine the expected value per turn for flipping nn size 12{n} {} times.

Progression

Students have another opportunity to analyze how the expected value increases or decreases with the number of flips per turn in a game similar to Pig.

Approximate Time

5 minutes for introduction

15 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

Clarify the rules for this game. Without simulating or even playing the game, students can move directly to considering a specific number of coin flips and calculating the expected value per turn. For each case, it will be helpful for them to draw a diagram or make a list. To develop a generalization, they will want to organize their expected value data and look for a pattern.

Discussing and Debriefing the Activity

Be sure everyone understands that for Question 1, following the one-flip strategy will score 1 point half the time and 0 points half the time, for an expected value per turn of 1212 size 12{ { {1} over {2} } } {}.

The key to Question 2 is recognizing that the only way to get a nonzero score is to get two heads in a row, and that this happens 1414 size 12{ { {1} over {4} } } {} of the time, for a score of 2 points each time it happens. To help students see this, you might ask, How many points do you get when you get a nonzero score? Based on that probability, students should be able to see, perhaps using the “large number of games” approach, that the expected value per turn is again 1212 size 12{ { {1} over {2} } } {}.

For Question 3, you might need to focus the discussion on two ideas. What is the probability of getting a nonzero score? Students might have used an area model to see that the probability of getting three heads in a row is . How many points do you get when you get a nonzero score? From the fact that this outcome scores 3 points, students should see that the average score per turn is 3838 size 12{ { {3} over {8} } } {}.

Discussion of the generalization (Question 4) is optional. For a strategy of flipping nn size 12{n} {} times, the only way to get a nonzero score is to flip nn size 12{n} {} heads. The probability of doing this is 12n12n size 12{ { {1} over {2 rSup { size 8{n} } } } } {}, and the score for a sequence of nn size 12{n} {} heads is nn size 12{n} {} . Thus the expected value per turn for an nn size 12{n} {}-flip strategy is n2nn2n size 12{ { {n} over {2 rSup { size 8{n} } } } } {}, which is 1212 size 12{ { {1} over {2} } } {} if n=1n=1 size 12{n=1} {} or n=2n=2 size 12{n=2} {} and then decreases as nn size 12{n} {} increases.

Key Questions

Is there a convenient way to keep track of the expected values for nn size 12{n} {} flips? If you use a table, what labels might you use?

Can you see a pattern to help you generalize the expected value per turn for an nn size 12{n} {} -flip strategy?

Supplemental Activities

Pig Tails Decision (extension) poses a strategy question for the game of Pig Tails in preparation for similar questions about Little Pig and Big Pig that students will soon encounter.

Get a Head! (extension) poses problems involving repeated flips of a coin and asks students to apply their developing understanding of probability and expected value. Question 2 asks them to analyze a game that could be arbitrarily long.