CHUCK-A-LUCK WORKSHEET



Name Partial Solutions / Examples

Date ___________

CHUCK-A-LUCK WORKSHEET (Answers will vary! This is an example.)

1) Using intuition only, is the game of Chuck-a-Luck fair? YES or NO Why or why not?

|Answers will vary! This is an opinion and should not be counted right or wrong. |

|The question is designed to get them thinking about “fairness” of a game. |

|I envision possible answers such as: Since you get to roll three dice, you have 3 of |

|the possible 6 outcomes covered so I would expect to win $1 about half the time |

|except when I match two or three times (pretty rare), so I would expect to win about |

|as much as I lose. |

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2) Record the results of CHUCKTWO.

These are sample results and will obviously vary for each student.

|Sim-# |Lost |One’s |Two’s |Three’s |Played |$-Lost |$-Won |Difference |

|1 |122 |80 |14 |0 |216 |-122 |108 |-14 |

|2 |130 |72 |14 |0 |216 |-130 |100 |-30 |

|3 |119 |86 |11 |0 |216 |-119 |108 |-11 |

|4 |129 |68 |19 |0 |216 |-129 |106 |-23 |

|5 |118 |77 |21 |0 |216 |-118 |119 |1 |

|6 |127 |70 |17 |2 |216 |-127 |110 |-17 |

|7 |126 |77 |12 |1 |216 |-126 |104 |-22 |

|8 |126 |71 |17 |2 |216 |-126 |111 |-15 |

|9 |124 |79 |11 |2 |216 |-124 |107 |-17 |

|10 |129 |74 |10 |3 |216 |-129 |103 |-26 |

|Totals |1250 |754 |146 |10 |2160 |-1250 |1076 |-174 |

3) Using the totals of your results in the previous table, complete the following experimental probabilities to the nearest hundredth of a percent.

Probability of

|Lose $1 |Win $1 |Win $2 |Win $3 |

|57.87% |34.91% |6.76% |0.46% |

4) How much did you win or lose per game in the total 2160 games you have data for? (Please calculate it to four decimal places.) Show how you calculated it! Then write in words what the numbers represent

|$174 / 2160 = |$.0805555555… = |$0.0806 |

|Total dollars lost divided by the total number of games played equals the loss per game to 4 decimal places. |

5) If you win or lose as much per game as you came up with in #4, how much would you expect to win or lose in 216 games? Show how you calculated your answer! Round your final answer to the nearest penny.

|(-$0.0806 * 216 games) = -$17.4096 = -$17.41 |

6) To find the Expected Payoff of a single game (Expected Value), you multiply each payoff times it probability of occurring. Using your experimental probabilities from the table in problem #3, compute the “experimental” expected value of Chuck-a-Luck.

|Outcome |Lose $1 |Win $1 |Win $2 |Win $3 |

|Probability |0.5787 |0.3491 |0.0676 |0.0046 |

|Payoff |-1 |1 |2 |3 |Total |

|Prob.*Payoff |-0.5787 |0.3491 |0.1352 |0.0139 |-.0805 |

7) Was your answer to #6 the same (or very close) to your answer for #4 ?

YES or NO

8) Should they be the same?

YES or NO

9) If they are very close, but not identical; explain why.

|Answers will vary! Explanation should include a statement about rounding or |

|Round-off error. |

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10) Now that you know the “experimental” Expected Value of the game of Chuck-a-Luck (problem #6), would you expect to win in the long run or lose in the long run? Mathematize your answer by commenting on how much you would expect to win or lose in 10,000 games.

|Answers will vary! I would expect to lose in the long run! Since my |

|experimental expected value was -$0.0806, I would expect to lose about $806 if I |

|played 10,000 games. |

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11) Using what you know about the Binomial Theorem as applied to Probability (restated below for reference), compute the following theoretical probabilities: (Express your answers in fractional form, decimal form, and a percent to 2 decimal places)

|The Binomial Theorem (applied to Probability) |

|Suppose an experiment consists of a sequence of n repeated independent trials, each trial having two possible outcomes, A or |

|not A. If on each trial, P(A) = (p) and |

|P(not A) = (1-p), then the binomial expansion of [p + (1-p)]n, |

|{ nCn*pn + … +nCk*pk*(1-p)(n-k) + … + nCo*(1-p)n }, gives the following probabilities|

|for the number of occurrences of A: |

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|Outcome |

|n A’s |

|… |

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|k A’s |

|… |

|0 A’s |

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|Probability |

|nCn *pn |

|… |

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|nCk *(p)k*(1-p)(n-k) |

|…. |

|nCo *(1-p)n |

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Show your computations as in the following example.

Example: The probability of 1 die matching my chosen number and the other two not matching is:

3C1*(1/6)1*(5/6)2

= 3 * (1/6) * (25/36)

= 3 * (25/216)

=75/216 (fractional form)

=.3472222222… (decimal form)

|Probability |No match (Lose $1) |One match (Win $1) |Two match |Three match (Win $3) |

| | | |(Win $2) | |

|of | | | | |

|Fractional Form |125 / 216 |75 / 216 |15 / 216 |1 / 216 |

|Decimal Form |.5787037037… |.3472222222… |.069444444444… |0046296296… |

|Percent |57.87% |34.72% |6.94% |0.46% |

Space to show your work!

3C0*(1/6)0 * (5/6)3 =

= 1 * 1 * (125 / 216)

= 1 * (125 / 216)

=125 / 216 (fractional form)

=.5787037037… (decimal form )

3C2*(1/6)2 * (5/6)1 =

= 3 * (1/36) * (5 / 6)

= 3 * (5 / 216)

=15 / 216 (fractional form)

=..069444444444… (decimal form )

3C3*(1/6)3*(5/6)0 =

= 1 * (1 / 216)*(1)

= 1 * (1 / 216)

=1 / 216 (fractional form)

=.0046296296… (decimal form )

12) To find the Expected Payoff of a single game (Expected Value), you multiply each payoff times it probability of occurring. Using the theoretical probabilities from problem 11, compute the “theoretical” expected value of Chuck-a-Luck. Hint! Leave your probabilities as fractions.

|Outcome |Lose $1 |Win $1 |Win $2 |Win $3 |

|Probability |125 / 216 |75 / 216 |15 / 216 |1 / 216 |

|Payoff |-1 |1 |2 |3 |Total |

|Prob.*Payoff |-125 /216 |75 / 216 |30 / 216 |3 / 216 |-17 / 216 |

| | | | | |-.0787037037… |

13) Using your expected value per game from #12, how much would you expect to win or lose in 216 games? Show how you calculated your answer! Round your final answer to the nearest penny.

(-$17 / 216) * 216 games = -$17

14) Explain what you would do to make this game “fair” and justify your answer mathematically by using an expected value table.

|Answers may vary! Since you can not change the probability of a fair die, the |

|answer should include changing the payouts for winning. The easy way to make it |

|fair is change winning $3 to winning $20. This will change your expected value |

|(see #12) to $0 and the game will be fair! The mathematical justification should |

|include an updated “Expected Value table” (#12) with an expected value of $0. |

15) Compare the theoretical expected value you computed in #12 with the experimental expected value you computed in #6. Comment on whether or not they were different, how different, were you surprised, etcetera.

|Answers will vary! |

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16) How would you change this game so it could be used at a carnival to raise money for charity? It should have at least one big payoff to entice people to play, but it has to be unfair in order to raise money for charity. Let’s say the game should have about a 90% payback. For a 90% payback, 10% of the money that was gambled goes to the charity, and 90% goes to the player(s). Hint!!! Your expected value should be . . . Justify your answer mathematically.

|Answers will vary! By adjusting the payback(s) they should come up with an |

|expected value of negative $0.10. One way to do this is to set the paybacks as |

|Match once (win $1), match twice (still win $1), match three times (win $13.40). |

|There are many other possible answers. |

|The mathematical justification should include an updated “Expected Value table” |

|(#12) with an expected value of negative $0.10. |

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