Transcript Preparing for Success in Algebra KICK

Preparing for Success in Algebra
English Language Learners in Mathematics
A Collaboration between:
 Los Angeles USD
 University of California, San Diego
 San Diego State University
 University of California, Irvine
Chuck-a-Luck

Problem Scenario: The game of Chuck-aLuck is an old carnival game which we wish
to analyze in this activity. You’ll have an
opportunity to simulate the game and
delve into its mathematics.
The “player” simply chooses a number from
1 to 6 and bets on it. Three dice are rolled
and if the chosen number appears on one
die, the player gets his/her money back
plus the same amount. If two dice show
the chosen number, then the player gets
his/her original money back plus twice the
amount. Finally, if all three dice show the
chosen number then the player gets
his/her original amount plus three times
the amount.

For example, if I bet $1 on the number 5,
and after the roll of the three dice two 5’s
show heads-up, I get my $1 back plus $2
more. It is pretty simple to play and to
simulate in a classroom. Here are your
tasks.
Procedure
Please rearrange yourselves at your tables
as necessary so there are 8 people at each
table and create partners so there can be 4
simultaneous instances of the game at
each table.
Select one person in the partnership to be
the carnival employee who rolls the dice
and the other the carnival attendee who
selects the number and wagers $1 on each
roll.
Play the game with your partner 25 times
with the carnival employee keeping track
of the attendee's wins and losses on the
accompanying table.
Create a total of 100 instances by
combining the results of all 4 groups at
your table. If the player is ahead by $7 for
example, express that as +7. If the carnival
is ahead by $7, express that as -7.
Combining the Results
We will then combine the results of all the
tables which will simulate several hundred
instances of the game. We would expect
the results from so many instances to be
close to the theoretical results which we
would like to analyze.
Instance
Win 1$
Win 2$
Win 3$
Lose 1$
…………..
…………….
…………….
……………
1
2
3
……….
24
25
Totals
times 1
times 2
times 3
times -1
Sample Space
The sample space is an exhaustive list of all
the possible outcomes of an experiment.
Each possible result of such a study is
represented by one and only one point in
the sample space. For ease of calculation,
we have three dice of different colors so
they can be distinguished. What is the
sample space for this experiment?
An Event

An event E is any collection of outcomes of
an experiment. Formally, any subset of the
sample space can represent an event. . For
example if we are considering the event E
of obtaining exactly one 5, then the
number of outcomes that have exactly one
5 form the event. The possible number of
5's for any outcome are 0,1,2,or 3.
Independent Events
Independent events are events such that
the outcome of one event has no bearing
on the outcome of the other event e.g. if
we roll two dice, the value on one of them
is unaffected by the value on the other.
Probability
Probability provides a quantitative
description of the likely occurrence of a
particular event. Probability is
conventionally expressed on a scale from 0
to 1; a rare event has a probability close to
0, a very common event has a probability
close to 1.
Equally likely outcomes
In some experiments, all outcomes are
equally likely. In our sample space above,
not all outcomes are equally likely. In a
raffle, all raffle ticket holders are equally
likely to win, that is, they have the same
probability of their ticket being chosen.
Mutually Exclusive
Two events are mutually exclusive if both
cannot occur simultaneously. Getting an
odd or an even number on the roll of a die
are mutually exclusive events. Getting an
odd number and getting a number greater
than 3 are not. We add probabilities to get
the probability of one mutually exclusive
event or the other.
An Example
For example, if one event is obtaining a 1
or a 2 on one roll of a die, the probability
of this event is 1/3. A second event could
be getting a 5 or 6 which also has
probability 1/3. The probability of getting
one or the other of these mutually
exclusive events is 1/3+1/3 = 2/3.
Independent Events

We obtain the probability of two
independent events both occurring by
multiplying their respective probabilities.
For example, if we roll two dice, the
probability of obtaining a one on both dice
is calculated as 1/6 times 1/6 or 1/36.
Expected Value
The expected value, EV is the result that
should occur if we perform the experiment
for an extended period of time but it is
usually expressed in terms of a single
instance which may or may not be possible
to achieve.
The Expected Value for Chuck-a-Luck
EV = -$1 (125/216) + $1 (75/216) + $2 (15/216) +
$3 (1/216) = -$(17/216) ≈ -$0.08
On average, you should expect to lose
approximately $0.08 each time you play a round
of the game. In 100 games, one should expect to
lose $8.
A Fair Game
A fair game is a game where the expected
value is 0 i.e. you expect to be even after
an extended period of time playing the
game. The Chuck-a-Luck Game is not fair
as we have seen.
Making it Fair
How might the rules of the game be
changed so that it is fair? Can you see how
to do it by adjusting the payouts for two
5's and/or three 5's?