The Lottery Question
The following work will be assessed using MYP Criterion A (Knowledge and Understanding) and Criterion D
(Reflection in Mathematics).
Lotteries are held throughout the world and are used to raise funds for various causes. For example, a portion of the
money collected for the California Lottery goes to fund the California public school system. However, most people
who purchase lottery tickets do so with the hope of winning A LOT of money!
Here is how a typical lottery might work:
A player purchases a ticket for $1 and chooses six numbers from 1 – 50. Six numbers are then chosen at random
from a container with the numbers 1 – 50 inside. If the player’s six numbers match the winning six numbers, she is
the $8 million Jackpot Winner! It is important to note that the order in which the numbers are drawn out of the
container does not matter.
Assume that you have purchased a ticket in the UNIS Lottery, similar to the lottery described above. When playing
the UNIS Lottery, you can win in the following ways:
Match
All 6 numbers
5 of 6 numbers
4 of 6 numbers
3 of 6 numbers
Win
$8,000,000
$4,000
$200
$10
Criterion A – Knowledge and Understanding
1. Find the Probability of winning each prize ($10, $200, $4,000, $8,000,000) and the probability of losing.
2. Calculate the expected return
* Be sure to show all of the work you have done for both questions. It may be helpful to leave your working
(and your answers!) in fractional form with the same denominators.
Criterion D – Reflection in Mathematics
1. Approximate each of the five probabilities you found above by writing them as a fraction with a numerator of
1.
2. Use these approximate answers to provide real-world examples that have a similar probability of success.
*You may have to do some research to make sure that your examples are realistic and reasonably
accurate. Make sure that your examples clearly show that you understand the meanings of the
probabilities that you are discussing. Be sure to acknowledge any sources that you use.
3. Use your expected return to analyse the fairness of this lottery. If it is fair, explain why it is. If it is not fair,
how can the winnings (and not the structure of the game) be changed to make the game fair? Be specific
and show evidence!
4. Use a different method from the one you used earlier, to calculate the probability of winning the $8 million
Jackpot (ie the probability of getting all 6 numbers). Explain clearly why your alternative method works.
There are 6 numbers and 50 balls to start off with when playing the UNIS lottery. The formula being used in
the first part of the investigation is:
This formula is used to find the probability of winning each prize ($8,000,000, $4,000, $200, $10). Each term
in this formula represents different numbers. W represents the winning numbers, M representing my
numbers and N represents the numbers not selected.
To win $8,000,000 the player need to get 6 matching numbers out of 6 numbers. Substituting appropriate
numbers into this formula, the probability of winning $8,000,000 will look like:
To win $4,000 the player need to get 5 matching numbers out of 6 numbers.
To win $200 the player need to get 4 matching numbers out of 6 numbers.
To win $10 the player need to get 4 matching numbers out of 6 numbers.
The total of the probability of winning each prize is:
Still using the same formula, the probability of losing $1 is:
According to the tables above, it is almost impossible to win $8,000,000 than the other prizes ($4,000,
$200, $10) because the probability of choosing 6 matching numbers is lower than selecting 1 or 2 matching
numbers out of 6 numbers. In the other hand, there is a high possibility of losing $1 everytime you play the
UNIS Lottery. As seen above, there is 94.83% of losing $1.
The next part of the investigation is to find the expected return of the UNIS Lottery. The formula used to
solve this is:
In this case, the amount of money would be the prizes ($8,000,000, $4,000, $200, $10).
Expected return for each prize is:
Then if I add the total of these results, I would get the expected return of the UNIS Lottery.
I will be approximating the five probabilities I found above as a fraction with a numerator of 1.
Approximation of the probability of winning:
$8,000,000:
$4,000:
$200:
$10:
Approximation of the probability of losing $1
There are real life examples that have similar probabilities to the 5 probabilities above.
Probability
Real Life Application
The odds f becoming president
The odds of tsunami killing a local coastal citizen
The odds of dying in fire or smoke
The odds of dying from heart disease
The information contained in this table above came from:
http://www.livescience.com/3780-odds-dying.html
Using the expected return of the UNIS Lottery which was:
This shows that this lottery isn’t fair because the probability of the player losing money is high when the
probability of winning money is low. There is a way to make this lottery a fairer game.
The number of the numbers the player have to choose from could be decreased. So in this case there were
50 numbers in total to select from. For example, instead of 50 numbers, the UNIS Lottery can have 40
numbers to select from. The formula that will be used when the UNIS Lottery is modified to become a
fairer game is:
Using this formula, the probability of winning the:
$8,000,000:
$4,000:
$200:
$10:
The total of the probability of winning each prize is:
The total of the probability of losing $1 is:
Since there is higher probability of winning each prize and lower possibility of losing $1. Therefore, the
game has become fairer to the player.
The method I used is not the only formula I can use to calculate the probability of winning the $8 million
Jackpot. There is another formula I can use, and they are:
Using this formula, the probability of winning the $8 million Jackpot is:
My alternative methods work because the answer is the same when using the method above,
Works Cited
Britt, Robert Roy. "The Odds of Dying." Live Science. TechMediaNetwork.com, n.d. Web. 15 Mar. 2013.
<http://www.livescience.com/3780-odds-dying.html>.
Criterion A – Knowledge and Understanding
Level of
Achievement
0
1-2
Descriptor
The student does not reach a standard described by
any of the descriptors given below.
The student attempts to make deductions when
solving simple problems in familiar contexts.
Indications
I do not reach a standard described by any of the
descriptors given below.
I have tried to answer most of the questions but I
am able not able to find any of the correct
probabilities.
3-4
5-6
7-8
Self
Assessment
The student sometimes makes appropriate
deductions when solving simple and more-complex
problems in familiar contexts.
The student generally makes appropriate deductions
when solving challenging problems in a variety of
familiar contexts.
The student consistently makes appropriate
deductions when solving challenging problems in a
variety of contexts including unfamiliar situations.
Comment:
Criterion D – Reflection and Evaluation
Level of
Descriptor
Achievement
0
1-2
3-4
5-6
Self
Assessment
I am able to find the probability of winning the
Jackpot and make progress in solving the other
questions.
I am able to find many of the probabilities
correctly. I attempt to use the formula to
calculate the expected return of the game.
I am able to find most of the probabilities correctly
and make only minor errors in the others. I am
able to calculate the expected return of the game.
The student does not reach a standard described
by any of the descriptors given below.
The student attempts to explain whether his or
her results make sense in the context of the
problem. The student attempts to describe the
importance of his or her findings in connection
to real life.
The student correctly but briefly explains
whether his or her results make sense in the
context of the problem and describes the
importance of his or her findings in connection
to real life. The student attempts to justify the
degree of accuracy of his or her results where
appropriate.
The student critically explains whether his or
her results make sense in the context of the
problem and provides a detailed explanation of
the importance of his or her findings in
connection to real life. The student justifies the
degree of accuracy of his or her results where
appropriate. The student suggests
improvements to the method when necessary.
Comment:
Indications
I do not reach a standard described by any of the
descriptors given below.
I have tried to illustrate the probabilities of winning
using non-mathematical examples. I attempt to
analyze the fairness of the lottery using expected return.
I have given appropriate examples to illustrate the
probabilities of winning. I analyze the fairness of the
lottery, with my conclusion being consistent with my
findings.
I have given detailed and accurate examples to
illustrate the probabilities of winning the lottery. I have
used my expected return calculation to analyze the
fairness of the game and have given an appropriate
suggestion to make it fair. I have given an alternative
method of calculating the probability of winning the
Jackpot and have explained it clearly.