The Lottery QuestionThe 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 WinAll 6 numbers $8,000,0005 of 6 numbers $4,0004 of 6 numbers $2003 of 6 numbers $10

Criterion A – Knowledge and Understanding1. 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 Mathematics1. 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 ApplicationThe 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 Descriptor Indications

0 The student does not reach a standard described by any of the descriptors given below.

I do not reach a standard described by any of the descriptors given below.

1-2The student attempts to make deductions when solving simple problems in familiar contexts.

I have tried to answer most of the questions but I am able not able to find any of the correct probabilities.

3-4 The student sometimes makes appropriate deductions when solving simple and more-complex

I am able to find the probability of winning the Jackpot and make progress in solving the other

problems in familiar contexts. questions.

5-6The student generally makes appropriate deductions when solving challenging problems in a variety of familiar contexts.

I am able to find many of the probabilities correctly. I attempt to use the formula to calculate the expected return of the game.

7-8 The student consistently makes appropriate deductions when solving challenging problems in a variety of contexts including unfamiliar situations.

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.

Self Assessment

Comment:

Criterion D – Reflection and EvaluationLevel of

Achievement Descriptor Indications

0 The student does not reach a standard described by any of the descriptors given below.

I do not reach a standard described by any of the descriptors given below.

1-2

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.

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.

3-4

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.

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.

5-6 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.

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.

Self Assessment

Comment: