Download CHUCK-A-LUCK WORKSHEET

Teacher Material …. Partial Solutions / Examples …. Do Not give to Students
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?
Why or why not?
YES
or
NO
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.
2) Record the results of CHUCKTWO.
These are sample results and will obviously vary for each student.
Sim-#
1
2
3
4
5
6
7
8
9
10
Totals
Lost
122
130
119
129
118
127
126
126
124
129
1250
One’s
80
72
86
68
77
70
77
71
79
74
754
Two’s
14
14
11
19
21
17
12
17
11
10
146
Three’s
0
0
0
0
0
2
1
2
2
3
10
Played
216
216
216
216
216
216
216
216
216
216
2160
$-Lost
-122
-130
-119
-129
-118
-127
-126
-126
-124
-129
-1250
$-Won
108
100
108
106
119
110
104
111
107
103
1076
Difference
-14
-30
-11
-23
1
-17
-22
-15
-17
-26
-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
57.87%
Win $1
34.91%
Win $2
6.76%
1
Win $3
0.46%
Teacher Material …. Partial Solutions / Examples …. Do Not give to Students
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
Payoff
Prob.*Payoff
0.5787
-1
-0.5787
0.3491
1
0.3491
0.0676
2
0.1352
0.0046
3
0.0139
Total
-.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.
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
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Teacher Material …. Partial Solutions / Examples …. Do Not give to Students
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.
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
n
P(not A) = (1-p), then the binomial expansion of [p + (1-p)] ,
n
k
(n-k)
n
{ nCn*p + … +nCk*p *(1-p)
+ … + nCo*(1-p) },
gives the following probabilities for the number of occurrences of A:
Outcome
Probability
n A’s
…
n
…
nCn *p
k A’s
k
(n-k)
nCk *(p) *(1-p)
…
0 A’s
….
n
nCo *(1-p)
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:
1
2
3C1*(1/6) *(5/6)
= 3 * (1/6) * (25/36)
= 3 * (25/216)
=75/216 (fractional form)
=.3472222222… (decimal form)
Probability No match
One match
Two match
Three match
(Lose $1)
(Win $1)
(Win $2)
(Win $3)
of
75 / 216
Fractional 125 / 216
15 / 216
1 / 216
Form
Decimal
.5787037037… .3472222222… .069444444444… 0046296296…
Form
Percent
57.87%
34.72%
6.94%
0.46%
Space to show your work!
0
3C0*(1/6) *
(5/6)3 =
3
Teacher Material …. Partial Solutions / Examples …. Do Not give to Students
= 1 * 1 * (125 / 216)
= 1 * (125 / 216)
=125 / 216
(fractional form)
=.5787037037… (decimal form )
2
3C2*(1/6)
* (5/6)1 =
= 3 * (1/36) * (5 / 6)
= 3 * (5 / 216)
=15 / 216
(fractional form)
=..069444444444… (decimal form )
3
0
3C3*(1/6) *(5/6)
=
= 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
Payoff
Prob.*Payoff
125 / 216
-1
-125 /216
75 / 216
1
75 / 216
15 / 216
2
30 / 216
1 / 216
3
3 / 216
Total
-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.
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Teacher Material …. Partial Solutions / Examples …. Do Not give to Students
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!
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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