Explorablers

ROCK PAPER SCISSORS
an
EXPLORABLE CLUSTERS FOR PROBABILITY CONCEPTS

F Richard Singer III www.conceptualstudy.org

Overview: This paper contains the specific example of an explorable cluster on elementary probability concepts that is taken from the paper Explorable Clusters. I want to emphasize that an explorable cluster is merely one type of resource that can be designed for constructivist learning. It is a type that can be used in an extremely flexible an open-ended manner by a mentor to both assist students in immediate concept construction and provides a basis for the construction of fairly remote concepts at a later time. It is also a type of resource that easily lends itself to being redesigned.

Acknowledgement: This explorable cluster was suggested by an activity from a session given by Julie Dangel at the 2004 ACT conference.

The RPS Explorable Cluster: Read this paradigm instance in details if it seems interesting. Otherwise skimmed it for its form. The initial realm of exploration centers on the Rock-Paper-Scissors social practice. The main target for RPS is any group of six or more students who are beginning to learn about probability and who understand that rock is covered by paper, paper is cut by scissors, rock breaks scissors. They also know how to implemented this between two persons.

Activity 1 This activity is an outguessing competition. It involves several pairs of persons. For each pair, call the competitors A and B. A game is 3 sessions, with each session consisting of 5 decisive trials. A trial is decisive if A and B make different responses. Results are recorded for each session. Winning a game means winning more sessions than the opponent. A pair may engage in as many games as they choose.

Considerations: This activity is followed by discussions involving some of the following considerations. Feel free to go beyond these or imagine other considerations. If you think of others then you may want to save them for further explorations.

(1) Relate winning a session to luck, skill, a combination of luck and skill, something else. Imagine that B is a computer playing at random. Imagine that B never does the same thing twice in a row.

(2) If a session involves pure luck, do you find it plausible that A could win by 5 to 0? Compare this to what actually happened. Is it at all likely that A could win all 3 sessions 5 to 0? Would this be likely if every person in the world entered this competition?

(3) One way for A to win a session is AABBA, i.e. A wins the first two, B the next two, A the last. List all ways for A to win by 3 to 2. How many ways are there for B to win by 4 to 1? Of the ways a session might turn out, what fraction of them are 3 to 2 wins by A. Continue this type of analysis.

(4) Discuss the meanings of likely and probable. Formulate a numerical concept of probability and relate this to winning a session 5 to 0.

Activity 2 This activity involves several triples of persons. For each triple, call the competitors A and B and C. A gets a point if all responses are the same. B gets a point if all responses are different. Otherwise C gets a point. There is a single session of 27 trials for each triple. A record of winners is recorded as the session proceeds. The winner is the one who gets the most points.

Considerations: Discuss Activity 2, in terms of some of ideas considered for Activity 1. Also discuss your concept of fairness and how it would apply to these competitions. Discuss modifications that you might make in relation to the concept of fairness.

Further Exploration: Exploration of this realm can be terminated at any time or it can be extended using additional considerations like those below or those suggested by students. It can also be extended if student formulate additional activities.

(1) A never does the same thing twice in a row in Activity 1. Find a strategy that enables B to win more than half of the time. Would it be fair for B to use this strategy?

(2) Suppose there are different prizes for winning in Activity 2.

(3) Discuss the relationship between the use of rock-paper-scissors as a competition and it usual role as a social practice. Relate this to flipping a coin.

Further Realms: A successor realms is a realm that can be explored at any time after exploring the initial realm. RPS includes four developed successor realms. More advanced resources for some these realms can be accessed from the website, as indicated in the supplement.

Coin flipping probabilities and simple voting probabilities with reasons for voting

Dice rolling and tack flipping probabilities

Plausibility concepts versus probability concepts

Social practices involving chance

There are several additional undeveloped successor realms that are suggested for exploration.

Public opinion poles

House profits from a various gambling devices

Probabilities and what actually happens

The use and misuse of probability statements and statistical claims

Designing strategies

Intended Educational Significance: What I am deliberately trying to do in designing this explorable cluster includes a number of significance considerations. With more thought others would occur and this might have resulted in a different design. For instance, altho statistical inference is related to probability, I gave no thought to this, and so this was not significant in designing this explorable cluster. Most of the significance could be described as laying a foundation for concept construction or enhancement. This includes laying a foundation for effectively using these concepts. Thus I use key concepts as a shorthand to indicate values of the significance parameter. Set notation indicates that I thought of these concepts primarily as a related set. The symbol ‘® ’ means that I thought of them as a chain.

{luck, skill, probability, plausibility} ® {fair competition, strategy} ® social practice
counting outcome ® {binomial theorem, binomial distribution} ® concept of probability ® combination and permutation ® more advanced probability applications such as poker
concept of probability ® equally likely outcomes ® sample spaces
improbable events ® {gamblers ruin, limitations of statistical conclusions}

Coin Flipping and Voting:

Activity If you flip a coin 6 times one way in which you might obtain the same number of heads and tails is HHTTHT, i.e. Heads twice, tails twice, heads, tails. Have each student estimate what they think is the probability of getting the same number of heads and tails. Have each student flip a coin 6 times, recording the results. Repeat this at least three more times, recording the results.

Considerations:

(1) Compare probability estimates to what actually happened. Determine the following probabilities when flipping a coin 6 times. Express them both as fractions and as decimals to the nearest thousandth.

exactly 1 head

exactly 3 heads

exactly 5 heads

an even number of heads

Compare these probabilities to what actually happened. Think of other probabilities involving 6 coins flips.

(2) A game consists of 8 sessions in which a coin is flipped 3 times. Player S gets 4 points if the result is either HHH or TTT. Otherwise Player D gets 1 point. Estimate the total number of points S might be expected to get. Do the same for D. Suggest a modification so they would expect to get about the same number of points.

(3) What is the probability of getting the same number of heads and tails when flipping a coin 2 times? What is the probability of getting the same number of heads and tails when flipping a coin 4 times? Estimate what the probability of getting the same number of heads and tails when flipping a coin 8 times. What about 10 times?

Story Bo’s club is voting at their meeting tonight for two candidate H and T who are running for club president. Altho Bo wants H to win, he has a chance to see a free football game instead of going to the meeting. He undecided about whether to vote. Bo knows that 10 voter are firmly committed to H and 10 voters are firmly committed to T. There are 6 other expected voters who are leaning equally towards H and T.

Considerations:

(1) Assuming each of the 6 undecided voters are equally likely to vote for H or for T, what is the probability that Bo’s vote could keep the election from ending in a tie?

(2) Suppose Bo finds out that there are only 3 undecided voters because the other 3 so called undecided voters have been convinced to vote for T. What is the probability that Bo’s vote could result in H winning the election.

(3) Again suppose that there are really 6 undecided voters, but that one of the firm supporters of H is too sick to go to the meeting and vote. What is the probability that Bo’s vote could make the election end in a tie? What is the probability that Bo’s vote could make H obtain win 14 to 12?

(4) Think about other possibilities. Can you think of any for which the probability that Bo’s vote would prevent T from winning would be greater than 1/3?

(5) In addition to wanting H to win, Bo has other reasons for voting for H. For instance the more votes H is able to obtain will enhance the influence H will have even if H looses. Think of a variety of other reasons that Bo might have for voting. Suppose Bo values these other reasons at about half as much as having H win. Could these other reasons actually carry more weight than wanting H to win?

Spinners and Tack Flipping: This realm involves probability outcomes that are not equally likely.

Activity 1. If you spin it 3 time you might obtain BRB. Have each student list all possible results, indicating what they think is the probability of getting the each of them might be. Have each student spin it 3 times, recording the results. Repeat this at least twice, recording the results.

Considerations:

(1) Compare probability estimates to what actually happened. Which is more probable BBB or RRR? Is there anyway to determine an exact probability for obtaining BBB and RRR?

(2) Assuming that the red region is exactly 1/3, determine the following probabilities when spinning the arrow 4 times. Express them both as fractions and as decimals to the nearest thousandth.

exactly 1 blue	exactly 3 blues	an even number of blues
exactly 1 red	exactly 3 reds	an even number of reds

Determine some other probabilities.

(3) A game consists of 27 spins. Player R gets 2 points for each red. Player B gets 1 point for each blue. Estimate the total number of points R might be expected to get. Do the same for D. Discuss the fairness of this game.

(4) A game consists of 27 sessions in which the spinner is used 3 times. Player R gets 2 points if the result has more reds. Otherwise Player B gets 1 point. Estimate the total number of points R might be expected to get. Do the same for D. Suggest a modification so they would expect to get about the same number of points.

Activity 2. If you flip a thumbtack you might get point down or point up.		D	U
If you flip a thumbtack 4 time you might obtain UDDU. Have students list all possible results, indicating what they think is the probability of getting the each might be. Have each student flip a thumbtack 4 times, recording the results. Repeat at least twice, recording the results.

Considerations:

(1) Compare probability estimates to what actually happened. Is there anyway to determine an exact probability for obtaining DDDD? What features of a thumbtack might have an effect on this probability?

(2) Suppose you have a thumbtack with probability 1/4 of obtaining of U on a single flip. Discuss probabilities of various outcomes for 4 flips

Further Exploration: Exploration of this realm can be terminated at any time or it can be extended using additional considerations suggested by students. It can also be extended if student formulate additional activities. One such consideration is to imagine a two state device with probability p of obtaining state 1 and ask about probabilities when using the device several times. Three state devices can also be considered.

Plausibility & Probability: Plausibility is a personal attitude towards some proposition, such as Jack will beat Jill in their next game of backgammon. Knowing that Jill has won over twice as many games as Jack, Ann finds this somewhat or fairly implausible, but not highly implausible. Not knowing anything about Jack and Jill, Bob has a totally open attitude about this proposition. Jack on the other hand, not being deterred by past experience finds it plausible. Jill, who does not feel as confident as her past experience and self knowledge should have suggested, finds it implausible but cautiously so. We can use numerical intervals to help indicate plausibility attitudes.

Highly Implausible: [1,9]		Somewhat Plausible: [51,69]
Fairly Implausible: [11,29]		Fairly Plausible: [71,89]
Somewhat Implausible: [31,49]		Highly Plausible: [91,99]

Other intervals may be used to indicate more about a plausibility attitude. [11,49] is a short way of saying at worst fairly implausible and at best somewhat implausible. [20,40] does the same but indicates less uncertainty, and is indicative of Ann’s attitude. [41,49] would indicative of Jill’s attitude, but after learning Ann’s attitude she revises her to [25,45]. This is more like Ann’s, but still leaning towards only somewhat implausible. [51,99] indicates Jack’s attitude of plausible, but otherwise uncommitted. [0,100] to indicates Bob’s attitude.

Note: The use of numbers is a device to assist in communication. No precise meaning need be assumed. Specifically these are not to be interpreted as probabilities. However, there is a way to make them precise, and this is done in the plausibility paper indicated in the supplement.

Do either Activity1a or Activity1b. Then do either Activity2a or Activity2b.

Activity 1a Consider the proposition that there is some form of life on Mars. In subgroups, have each student write down a plausibility interval indicating his/her attitude towards this proposition. Have them explain their attitudes to other members of their group. Have each write a revised attitude. Have them each write what they think would be an appropriate plausibility interval that represents a group consensus. The subgroups then report back to the whole group.

Activity 1b Separate into subgroups. Have each subgroup make up a proposition that they feel will evoke differing plausibility attitudes. Have each student write down a plausibility interval indicating an attitude towards this proposition. Have them explain these attitudes to other members of their group. Have each write a revised attitude. Have them each write what they think would be an appropriate plausibility interval that represents a group consensus. The subgroups then report back to the whole group.

Considerations:

(1) Consider the proposition F₁ that a fair coin that is to be flipped 5 times will come up heads each time. The probability that this will happen is 1/32. Explain why [3,4] might be a reasonable plausibility attitude to adopt towards F₁. Can you think of other plausibility attitudes that it would be reasonable to adopt towards F₁? Suppose replace F₁ by the proposition F₂ that a fair coin had been flipped 5 times and had come up heads each time. Does it make sense to talk about the probability of an event that may or may not have already happened?

(2) A proposition about a current or past state of affairs is either true or false. Why would it be appropriate to adopt a plausibility attitude towards it rather than a belief?

Activity 2a The mentor has a number x from {0,1,2,3,4,5}. Each student is handed a note. Four students S₁, S₂, S₃, S₄ obtain note with the special information indicated below. The other note merely says that some of the students have received some special information. The students with special information were selected at random, and the mentor gives no indication of who has any special information.

S₁: x ¹ 4.

S₂: If the x is even then x ¹ 2.

S₃: If the x is odd then x = 3.

S₄: x was chosen by flipping a coin 5 times and taking the number of heads.

Without any discussion, students write intervals indicating their initial plausibility attitudes towards the propositions that the mentor’s number is 3. The mentor displays these. Without discussion each student may make a revision and these are displayed by the mentor.

The students then explain these attitudes. S₁ and S₂ reveal the information they received, but S₃ and S₄ do not. Attitudes are revised again and displayed.

The students again explain these attitudes and S₃ and S₄ reveal the information they received, Attitudes are revised again and displayed.

Considerations: Did any of the plausibility attitudes shift very much? Did they shift towards or away from consensus? Given all the special information what can you say about the probability that the mentor’s number is 3. Relate this to plausibility attitudes.

Activity 2b Make up some activity which you think will demonstrate how plausibility attitudes can shift on the basis additional information and group discussion.

Established Social Practices and Chance:

Activity 1 Casting lots means making a decision by using lots (straws or pebbles etc.) that are thrown or drawn. Have each students consult a websites for information on casting lots. Some such sites are given below.

www.spirithome.com/paralots.html

www.annieshomepage.com/castinglots.html

www.jameslindlibrary.org/essays/casting_of_lots/casting.html

Considerations: Discuss social practice involving the use of chance. Begin with a focus on the information from the above websites or any others that were consulted. Discuss the pros and cons of others, such as using rock-papers-scissors to decide who goes first, coin flip in football games, lotteries for socially approved purposes. Relate casting lots to trial by combat.

Story Bo’s club prefers to settle all matters by discussions which reaches a consensus. When this is not possible they want to make sure that minority views have a chance of prevailing. If they cannot reach a consensus about an activity then each member places his choice in an urn and one is drawn at random. Instead of voting for a proposal in the usual manner each member of a club places a yes ball or a no ball in an urn. One of these is drawn at random to determine if the proposal passes.

Considerations: Relate this to probability. Discuss the pros and cons of such a social practice.