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Subject: Games for Teaching Probability #2: Relative frequency and histograms rss

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Pete K
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Recently I taught an introductory statistics course at a local community college. I was given a fairly detailed list of course objectives and the class textbook, which were previously selected by the Department, but I was otherwise given freedom to teach the course materials as I saw fit. One of the first things I planned to do was to introduce some in-class activities to hopefully liven up the lecture material and, in the parlance of the community college, promote an open and active learning environment. I mined my recent experiences with smaller hobby games in order to produce some learning experiences that are (hopefully) more engaging than the rudimentary examples of card-drawing, coin-flipping, etc. I gave each of these activities about 45 minutes to explain, demonstrate, give to the groups of students and follow up with a series of topical questions. This session report is the second of a planned series of six posts.

Games for Teaching Probability #2: Relative frequency and histograms

Relative frequency is a core concept in the study of statistics, and in my course was among the first probability formulas. If an outcome occurs f times out of n total events, the relative frequency of the event occurring is r = f/n.

I wanted to have a class activity where the students could get a hands-on feel for using relative frequency as a means for estimating the probability of a desired outcome. A game with a push-your-luck mechanism seemed ideal because of the decision making it forces on players. I produced several mock-ups of the classic dice game Can’t Stop, whose board design has the bonus feature of resembling a histogram of the combined outcome of a two-dice roll. Players use combinations of two dice (out of four rolled at a time) to advance their markers along tracks, which are the bars of the discrete “histogram.”



The goal of this game is to get one of the markers on the end of one of the tracks (1-12). On a player’s turn, he rolls four dice and makes two 2-dice combinations of his choosing. He then places a “runner” stone (he starts with three stones) at the beginning (first spot on the left) of the track corresponding to each 2-dice total. He then rolls again and makes combinations, placing the third stone on another track, or advancing a stone one space up the track.

For example, if he rolls 4-3-5-2 and already has a stone on the 7-track, he can make two combinations of 7 with the dice and advance the stone two spots. Between each roll, he can choose to stop and replace the stones with markers of his color (thus saving his progress for the next turn), or keep rolling. If he is unable to place a stone or advance a stone after a roll, his turn ends without moving any of his markers.

The rules were simple enough so that I could squeeze them on the printed 11 × 17 “board.” Players were either risk-averse, ending their turn early, or reckless enough to either win or go bust on any given turn.

Questions
1. The board of Can’t Stop is (with some adjustments) a large histogram. What is it a histogram of?

2. The length of the tracks in Can’t Stop are created to maintain a balance – the “2” track, though much shorter, is not necessarily easier to complete than the “7” track. Why is this?

3-4. HYPOTHETICAL CASE – You are the second player in Can’t Stop, and it is your first turn. The first player has advanced one marker on her turn, up 5 spaces on the “7” track. Your first roll of the four dice is: 1, 1, 6, 6.

3. One strategy is to group the dice 1-1 and 6-6, and place the runners on the “2” and “12” tracks. Why would this be a good idea?

4. The other option is to group the dice 1-6 (the other 1-6 would not be used), and place the runner on the “7” track. Why would these be a good idea?

5. Can’t Stop is a classic example of a push-your-luck game, like Yahtzee and craps (the casino game). I think Can’t Stop is much more interesting than Yahtzee, and we will get to craps and other casino-style dice games later in the course. Based your short experience with this game, is it better to be aggressive in Can’t Stop, or more careful?

Verdict
Out of all the in-class activities in the class, this may have been the most popular. In retrospect this should be hardly surprising, given the classic status of Sid Sackson’s 1980 design. The drawback to the activity was the amount of time required to get every playing correctly, but I had a relatively small class and was able to facilitate all of the groups. Most of the class also seemed to have a solid grasp of histograms and relative probabilities in the subsequent exam (and final), but this may have been due to my referring back to the Can’t Stop activity in other lectures.

Note: I realize that the Can’t Stop board, as well as my paper mock-up version, is not a correct representation of the probabilities associated with finding two-dice combinations out of four dice rolls. Others have done the math, which is easily attainable: I am leaving that information off of this article.

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Jim Cobb
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Alpharetta
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visit rollordont.com for a free computer game with a challenging AI player!
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visit rollordont.com for a free computer game with a challenging AI player!
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If you'd like to show your class a whole lot of probability in a short amount of time, you could point them to my game:

http://www.rollordont.com

It will show them a lot about the odds very quickly! :o)

Jim
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Pete K
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There's a list of all the sessions I did for probability demos here.
 
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