How long until we "Pig Out"? Part I

"What are the chances of a 'Pig Out'? Part I" is the most popular post on this blog. The activity ends by asking the reader to consider how many tosses it would take, on average, to roll a "Pig Out" and lose the points accumulated during the turn. Just in case you are not familiar with the game Pass the Pigs™, or didn't read the earlier post, here are the rules and a figure showing the scoring:

I developed the following problem solving workshop with my colleague, Dr. Mary Richardson, as part of a presentation for a Math in Action Conference.

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Understanding the Problem

In the dice rolling game, Pass the Pigs™, players are always on the look out for the dreaded "Pig Out." If it is tossed before a player passes the pigs to an opponent, the player loses all the points for the round.  A "Pig Out" occurs when the pigs land on opposite sides – dot and no dot (as seen above). Therefore, it would be good to know about how many tosses it typically takes before a "Pig Out" occurs.

Create a Plan

How might you solve this problem?

Please consider several possible solution methods

Carry Out a Plan

Try one of the plans you considered in the previous phase

Looking Back/Ahead

• Use your results to answer the question, "How many tosses would you expect it takes until a "Pig Out" is rolled?"
• Evaluate the plan you used (glows and grows) – What did you like? What would you do different?
• Describe an extension – What other questions arise from this game?

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Before I share our results, I want to provide you an opportunity to try solving the problem yourself. There is a free game here if you want to gather your own data. In future posts, I will describe two approaches we used to answer this important question.

What are the chances of a “Pig Out”? Part II

Last post, I shared with you the first half of an article I wrote with Dr. Mary Richardson (here is another article I wrote with Mary addressing fair games). Learners have just predicted the chances of rolling a "Pig Out" in the popular Pass the Pigs game. They are now ready to explore the experimental probability associated with this question.

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The Activity (continued)

After learners share their subjective estimates, they are ready to test them by developing an experiment, as described on the Activity Worksheet.  In the activity, learners create possible plans for determining an estimate of the chances of a “Pig Out”.  Once plans have been suggested, the class discusses the plans, agrees on a plan to utilize, and carries out the plan.  One possibility would be to have each pair of learner toss the pig dice several times in order to see how often a “Pig Out” occurs. For example, within each pair, the following tasks are assigned:  one member tosses the pigs; and the other member records whether or not the pigs landed in the “Pig Out” position.  After one learner completes ten trials, the students switch tasks and ten more trials are conducted.  This provides partners with a total of twenty trials that can be used in determining an experimental probability.

Once the learners have conducted the experiment, we ask them to refine their original guesses.  When asked how they could be more certain of the probabilities, most respond that a longer experiment (more tosses) would result in more certainty.  However, they recognize that it is unrealistic for each of us to toss the pigs a very large number of times.  Instead, we decide to use the results from the entire class to get a better estimate for the probability.  We draw a number line on the whiteboard.  Each learner writes his/her own result from their tosses on a sticky note and posts the note appropriately on the line in order to form a dot plot.  After we have discussed the data for the whole class, everyone makes a new estimate for the probability.

Learners are asked to share their new probability estimates.  Most learners select the class median or class mode as their new estimate.  We estimate that a “Pig Out” is rolled one time in every five rolls.  Of course, differences in desktops, and the very nature of the experiment may produce varying results.

Conclusion

The important concept in this activity is that one cannot simply count outcomes for every object and assign a theoretical probability to any given outcome.  By exploring objects that cannot be assigned theoretical probabilities, learners seem to have a greater understanding of when a theoretical probability can be assigned.  Furthermore, learners recognize the need to generate data in order to obtain experimental probabilities

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The exploration did not end there for us. We found several websites dedicated to Pass the Pigs. They contained online versions of the game and data collected in secondary classrooms. We used the data to update our experimental probabilities and look at new questions.

Sadly, the websites we found are no longer, but Wikipedia has data on rolling a single pig 11,954 times. Maybe you could use these to collect your own data to answer your own questions. For example, in order to win the game, you want to know when to pass the pigs. Therefore, the next question we asked was, "How long until we "Pig Out"? We came up with two ways to answer this question but I bet there are more.

What are the chances of a “Pig Out”? Part I

Several years ago, I wrote the following article with my colleague, Dr. Mary Richardson, for MCTM's Mathematics in Michigan (Volume 43, Number 1). Given that the issue is no longer available, I thought it would be appropriate to share it here. Because of the article's length it has been edited and will be split between two posts.

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Introduction

Middle grades learners should be provided ample opportunities to explore probability and statistics topics.  In particular, learners should be encouraged to collect, organize, and describe data, summarize the data, make predictions and conclusions based on the data, and test these predictions and conclusions (NCTM 2000).  [Update: this connects to CCSS 7.SP.C.6.] In the following activity, learners use the Pass the Pigs dice game to design a simulation experiment in order to estimate a probability. 

The Game

Pass the Pigs involves two tiny rubber pigs that players throw like dice (rules). Each pig can land in six different ways resulting in various point values.  A turn consists of a player taking as many rolls as he or she dares until: (1) deciding to stop and recording the total score for that turn, (2) a “Pig Out” is rolled and the player records a score of zero for that turn, or (3) an “Oinker” is thrown and all points accumulated in the game thus far by the player are lost.

The Activity

In this activity, learners explore the experimental probability of obtaining a “Pig Out” in the Pass the Pigs dice game.  The activity works best if learners have experience calculating measures of center, including mean, median, and mode.  Learners work in groups of two.  Each pair has one Pass the Pigs dice game and a flat table or desktop on which to work.  Each learner has a sticky note and a copy of the Activity Worksheet.

To begin the lesson, each pair of learners examines their pig dice.  We discuss the possible outcomes if the pig dice are simultaneously tossed.  Each learner is then asked to estimate the probability that a “Pig Out” will occur when playing Pass the Pigs.  Learners should write down and share their estimates and the reasoning behind them.  This leads into a discussion about different types of probability.  In addition to the two types of probability traditionally targeted in middle school (theoretical and experimental), this activity provides the opportunity to introduce learners to subjective probability.  Learners recognize that this is a situation for which there is no easy way to determine a theoretical probability and no data on hand for them to use to calculate an experimental probability.  Therefore, they must assign probabilities based upon what they think will happen (subjective probability).

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To be continued... In the meantime, please estimate what you think is the probability that a "Pig Out" will occur and share it in the comments. Thank you.