Investigation 4 Blog Post

1. Describe four probability situations that involve two actions. Describe the outcomes for these situations.
Situation 1: A six-sided die with 6 equally likely sections: 1, 2, 3, 4, 5, and 6 and a fair coin with only two fair possible outcomes: heads or tails.  Out of both these events both of the outcomes are equally likely of occurring. Overall, there are 12 possible outcomes, breaking this down-- there are only 6 outcomes for the die and 2 outcomes for the coin, so 6*2=12 possible outcomes. The 12 outcomes are:
1. (heads 1)
2. (heads 2)
3. (heads 3)
4. (heads 4)
5. (heads 5)
6. (heads 6)
7. (tails 1)
8. (tails 2)
9. (tails 3)
10. (tails 4)
11. (tails 5)
12. (tails 6)
Situation 2: There is a spinner with 4 equally likely sections: red, orange, yellow, and green. There is also a bag of marbles containing one blue, one purple, and one pink marble of all equally proportions. There are again 12 different possible outcomes-- breaking this down, there are 4 possible outcomes on the spinner and 3 possible outcomes in the bag, so 3*4=12 possible outcomes. The 12 outcomes are:
1. (red, blue)
2. (red, purple)
3. (red, pink)
4. (orange, blue)
5. (orange, purple)
6. (orange, pink)
7. (yellow, blue)
8. (yellow, purple)
9. (yellow, pink)
10. (green, blue)
11. (green, purple)
12. (green, pink)
Situation 3: There is a spinner with two equal parts: numbered 1 and 2. There is also a fair two sided coin that has 2 possible outcomes: heads or tails. Therefore are only 4 possible outcomes-- breaking this down, there are only two possible outcomes for the spinner and only two possible outcomes for the coin, so 2*2=4 possible outcomes. The 4 possible outcomes are:
1. (1, heads)
2. (1, tails)
3. (2, heads)
4. (2, tails)
Situation 4: There is a bag containing only 4 marbles with equally proportions: blue, yellow, green, and red. There is also a fair coin with only 2 possible outcomes: heads or tails. Therefore, there are only 8 possible outcomes-- breaking this down, there are only 4 possible outcomes in the bad filled with marbles and only two possible outcomes for the coin, therefore 4*2=8 possible outcomes. The 8 possible outcomes are:
1. (blue, heads)
2. (blue, tails)
3. (yellow, heads)
4. (yellow, tails)
5. (green, heads)
6. (green, tails)
7. (red, heads)
8. (red, tails)

2. You can use an area model or a simulation to determine the probability of a situation that involves two actions. Explain how each of these is used.
Area Model: Basically, you simply find the theoretical or experimental probability of an outcome happening. To make an area model you first start drawing a large square, preferably a 10x10 square on grid paper as it's easier to do the calculations needed later on. After that, you have to represent each outcome individually, on adjacent parts of the grid/square. You would then label these parts after the possible outcomes/sample spaces that could occur in your experiment. To represent the probabilities of your outcomes occurring this you would draw in the lines needed. In an area model, dividing a box into 4 equal sections only shows that the outcomes are equally likely, but it's not likely that your outcomes in your situation will always be equally likely so, you should take into consideration the space of the sections within your area model as it is extremely crucial to the overall process. For example, if you were flipping a coin, the area model would look like this. Shown in the picture are the different outcomes possible. The "heads+tails" written on the top represent the first flip. The "heads+tails" written on the left side represents the second flip. You can see in the area model that all the boxes are the same size, which shows that you have an equal chance of landing on any of those combinations.
Simulation: A simulation is just a way to kind of copy or simulate your actual experiment so you could get an idea of all the outcomes. You could make a simulation digitally as the experiments are conducted quicker online and within an instant your results are created accurately. There are many websites that allow you to plug in the events in a situation and the parameters to make an easy simulation. Websites such as is really helpful and is easily useable:
http://media.pearsoncmg.com/curriculum/math/cmp3/math_tools/

3. Describe how you would calculate the expected value for a probability situation.
To find the expected value for a probability situation, you first have to find the theoretical probability of each possible outcome. After that, you have to place that outcome out of a certain amount, creating faction. It's best to have your fraction out of a 100 times because then the calculations will become easier. For example, suppose we are in a one-to-one free throw situation in basketball. The question that has to be asked is, how many points will a player be expected to score out of one hundred throws? If the basketball players had a free throw accuracy of 20%, the theoretical probabilities of scoring 0,1, and 2 points would be 80%, 16%, and 4%. So, that means that out of one hundred free-throws, this player would be expected to score 24 points as 80x0=0, 16x1=16, and 4x2=8, 0+16+8=24 points. However, to find the expected value, you need to figure out the amount of points that will be earned out of one throw. So, 24/100= 0.24/1. Out of 1 throw or situation, this player would be expected to score 0.24 points.

4. Expected value is sometimes called the long-term average. Explain why this makes sense. 
The term "expected value" is from the value that is expected to occur in the future, or over the long term. So, long term average is another way of saying the future, and the average is also a way of saying expected value, so they go hand in hand. Again, in many respects, long-term averages are exactly the same things as expected values as they are both terms that describe what will happen over a certain amount of times in the long-run. This is why calling expected values long-term averages makes complete sense. The expected value/long term average of an outcome will always stay the same. An average never changes unless you add another factor into the game. 

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Appositive

Absolute

Prepositional

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What Do You Expect Investigation 2 Reflection

1. Describe how you can find the theoretical probability of an outcome. Why is it called a theoretically probability?
Theoretical probability is used to find the likeliness of an outcome of the probability of an outcome, just like experimental probability. However, there are some critical differences between these two types of probability methods. While the experimental probability of an outcome is found by conducting an actual experiment, theoretical probability is found by observing a situation and using calculations to come up with the probability of an outcome. Theoretical probability does not need you to perform an experiment to find the likelihood of a desired outcome. For example, let's say that there are two coins, and you wanted to see the likelihood of the outcome to be a match (HH or TT). Instead of actually performing an experiment with a good amount of trials, you would analyze the situation an then use calculations to form a theory. Here is how you would execute the theoretical probability for the problem stated above. First, I know that the probability of getting a heads or a tails on one coin is P(H or T)=1/2. This would be the same for the other coin, as there are still two sides. Then, you would multiply the 1/2 twice as there are only two coins: 1/2 * 1/2= 1/4. So, there is a 1/4 chance that there will be a match. This is using theoretical probability. Overall, it's called theoretical probability because your conclusions and your later predictions are based off of your theory of a situation. And remember, something that is theoretical is only an idea, not an actual or physical thing. Something that is theoretical is only an idea.

2.          a. Suppose two people do an experiment to estimate the probability of an outcome. Will they get the same probabilities? Explain.
This is a bit of a tricky question. This depends on the actual experiment performed by the two people as well as the number of trials that were executed in the experiment itself. It is possible that both people get the same results, or get results close to each other's. For example, let's say that a girl name Anya and a girl named Riya were both going to toss a coin 100 times. The theoretical probability probability of it landing on either heads or tails: P(H or T)=1/2. There is a 50% chance that the coin could land on either side when tossing the coin. So, when flipping the coin, you would believe that the results would support exactly that. However, the experimental probability may show otherwise. It's not certain that out of 100 flips, there would be 50 heads and 50 tails. Although that is not likely, it is likely to get the experimental probability with a 50-50 chance of it landing on either heads or tails. It is also likely that the coin will land close to 50 heads and 50 tails. Going back to the original question, it is not 100% certain that the two students who perform the experiment will get the same results. Their conclusions and outcomes are all dependent on how the experiment itself is set up (how many trials are performed) and the way that the information is presented.

             b. Two people analyze a situation to find the theoretical probability of an outcome. Will they get the same probabilities? Explain.
These people should get the same theoretical probability. Theoretical probability is only based on what the probability should be in THEORY. There is no factor that could influence one's result to be different from the other's. However, as we know what happens in real life, people make errors in their calculations or just don't understand the concepts. Apart from those minor factors, there is no reason for the results of the theoretical probability to differ. So for example, if they are examining a spinner with four colors (red, green, yellow, and blue), all colors of equal size, then both of the people's their theoretical probability should end up the same. The theoretical probability of the spinner landing on yellow is 1/4. This theoretical probability will always stay the same, unlike experimental probability which can change.

             c. One person uses an experiment to estimate the probability of an outcome. Another person analyzes the situation to find the theoretical probability of the outcome. Will they get the same probabilities? Explain.
The experimental probability of an outcome is likely to be close to the theoretical probability of the outcome. If we use the same example from question 2a, the theoretical probability that Anya or Riya would land on heads would be 1/2 or 50/100. Yet , neither people actually got exactly 50/100. Each of their experimental probabilities were different. The results showed that when it came to Anya's set of data, there was a 53% chance that her coin would land on heads after being tossed. 53% is quite close to 50% but not exact. But, Riya had a 46% of her coin landing on heads after tossing it. 46% is not close to 50%, this shows that it is also likely that they will not get the same probabilities when finding the experimental and theoretical probabilities. But remember, it is likely that they will get probabilities close to one another. Sometimes the experimental probability can be affect by how you execute it and how may trials, this will change how accurate and closer to the theoretical probability the data is. Overall, it is quite unlikely that the people will get the exact same outcome when finding the theoretical and experimental probabilities.

3. What does it mean for a game to be fair?
For a game to be fair, there has to be an equal chance of winning and losing for each player. This doesn't mean that each player had to have a 50% chance of winning to be fair. it just means that each player has an EQUAL chance of winning. A game is also fair if there is a "fair" way of winning. To calculate this, you need to know the number of player that are playing. The, you would divide the full 200% evenly among the players. So, if there are 5 players, and the game is "fair", this would mean that each plater had a 20% chance of winning, which is fair. You can only do this when you know the the game is certainly "fair".

4. What is a sample space, and how can it be repeated?
A sample space is a list of all the possible outcomes from an experiment or any other given situation. This means that you have a list of all the possibilities written down. Generating a sample space can be extremely helpful when finding the likelihood of an outcome as you can see, right in front of you, all the possibilities. A sample space will help you get a deeper understanding of what the game could look like and show you if the game is "fair" or not. Generating a sample space is actually quite easy.One strategy when generating a sample space is drawing a tree diagram. The picture on the side an example of a tree diagram. This diagram shows the possible outcomes when tossing three coins. Another way to show a sample space is creating a matrix.

Design a Spinner

1. How did you decide how to design your spinner?

(The information provided above is written in a way called probability notation. Basically what P(red)=40% means it that the probability of the spinner landing on the red section is 40%.)

First: So what I did first was understand and decode the instructions that I already had...
1. Since 40% is just a fancy way of saying 4/10 or 40/100, I put four down on the row marked with the red square to the right (by using the blue up arrows). This then made the size of red "4" or "40%".

2. Then, since 20% is just a fancy way of saying 2/10 or 20/100, I put 2 down on the row marked with the blue square to the right (by using the blue up arrows). This then made the size of blue "2" or "20%".

3. Finally, I repeated the same process as I did with the red column (same logic and exact same actions) because in the criteria above the probability of both red and blue needed to be 40%.

The end result of sizing the colors looked like this...


Next step: I needed to figure out what the hard probability notations P(red or odd number)=100% and P(blue and 3)=20% meant. P(red or odd number)=100%... What I figured out from this is that all three parts needed to be odd numbers to ensure that there was at least a 100% chance that the spinner would land on either red or an odd number. P(blue and 3)=20%... I realized that because blue already took up 20% of the spinner, all I really had to do was label the section with the number "3". So by naming just random odd numbers off the top of my head, I decided to label red "1" and white "5":

1. I labeled the red section "1" as it is an odd number which supports the probability notation of P(red or odd number)=100%.

2. I labeled the blue section "3" as it is an odd number which supports the probability notation of P(blue and 3)=20%. I explain above that blue already takes up 20% of the spinner so the 20% would be labeled "3".

3. I labeled the white section "5" as it is an odd number which supports the probability notation of P(red or odd number)=100%.

In the end my final spinner looked like this...

2. How did you confirm that your spinner design met the criteria?
The criteria to be met:


I already know that the probability of the spinner landing on red and white is 40% each because they each take up 2/5 of the space on the face of the spinner. Therefore, the white and red sections of the spinner will have a higher chance of occurring than the blue section. However, this is what I expected, it is an inference that I inferred, but this may not necessarily happen. The improbable occurence would be if the spinner landed on the blye section more times than the other parts. This is improbable but not impossible. Secondly, for the probability of red or an odd number to be 100% landed on by the spinner, I realized that all the numbers on the spinner have to be odd in order for this to be a true statement, which they are. For the last parameter, I realized that blue is taking up 20% of the space, so all I had to do was label this section with the number "3"/ All of this shows that my spinner successfully meets all the criteria  required for this assignment. To test my assumptions and inferences, I ran an experiment, spinning the spinner 10,000 times:

My experiment results showed that the red and white sections were landed on at almost the same amount of times. It also shows that the blue section was landed on about half the amount of times the red and white landed on. 

3. Is there more than one acceptable spinner design?  If so, what will the spinners have in common? What will be different?
Yes, of course, mostly everything in the world has multiple solutions to it. As I said earlier on in this post, the number "3" as labeled on the blue section must stay blue in order to meet the criteria explained above. However, the other two number can be any odd number as the criteria doesn't specifically call upon certain number. For example: 
 
You could say that there will be an infinite amount of spinner as there are an infinite amount of odd numbers. But this is the only factor of the spinner's criteria that can vary. Otherwise, everything else stays the same and the proportions or sizes of the colors don't change and neither does anything else.

What Do You Expect Mathematical Reflections 1

1. How do you find the experimental probability that a particular result will occur? Why is it called the experimental probability?
          How: To find experimental probability you need to first perform the experiment at hand. Whether it's tossing coins or cups, flipping spoons, or even spinner a spinner, you have to first perform the experiment. For example, lets say that Mary wants to decide which desert to eat for dinner everyday of June-- either fruit, heads, or ice cream, tails. She does a trail to see what the probable outcome will be, tossing the coin 30 times. She needs to record the number of heads out of 30 and the number of tails out of 30. You would perform this experiment and record the results-- heads: 14 times and tails: 16 times. Secondly, you would right out the fraction/relative frequency using formal probability notation with the certain parameters that were chosen at the beginning of the experiment-- heads: 14/30 and tails: 16/30. Then, you would convert that fraction into a percentage, and after this, you can finally begin analyzing the data and begin making predictions about future outcomes.
          Why: The experimental probability, relative frequency, or also known as the empirical probability, is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment. Lastly, it's called experimental probability because the predictions and the outcomes are based off trials formed from an experiment.

2. In an experiment, are 30 trials as good as 500 trials to predict the chances of a result? Explain.
          NO... NEVER! In an experiment 30 trials is not as food as 500 trials to predict the chances of a result. The golden rule is that the more data, trails, and evidence you have, the better. When you are performing an experiment, you have to always try to have somewhat accurate results otherwise this doesn't help you in getting the information needed to solve your question or make further choices and predictions. Again, the more trials you have, the more accurate the results of you experiment will be as you have more and more data to work with. To predict the chances of a certain result through the experiment, you need to base that prediction off of good data that's somewhat reliable and can be almost accurate. A connection to a previous unit is the Are You Normal unit, the math teachers collect a huge sample size of all math students in a survey and then we analyzed the data. To do this experiment in the previous unit we need a big sample as the it was a very large survey and there weren't that many common answers. So to make a prediction on the population we needed a large sample, otherwise the data would be very hard to analyze. This idea of 500 trials being better than 30 trials directly relates to this previous unit.

3. What does it mean for the results to be equally likely?
          For results to be equally likely, they all have to have the same chances of occurring. It also means that the experiment has to be unbiased and fair. However, equally likely doesn't mean that results have a 50% and 50% chance. The most important part of determining if something is equally likely to another, you have to now how many parameters were set for the experiment. For example, if you have 4 different parameters or options, you would have to distribute these parameters or 100% evenly among these 4 parameters. Therefore, they would have the same chance of occurring, making none of them less likely to occur than another. Going with the example below each parameter or result would have  25% chance of occurring, and the results would be equally likely.

UN Day- Blog Reflection

1. Give a brief description of UN Day and the activities you undertook.

RLA: In RLA we had these two amazing poets come in and share with us some of their knowledge. They taught us quite a lot about themselves as writers and taught us some tips and advice when writing poetry. Then they taught us about P.I.P.E.S. which stands for projection, inflection, practice, eye contact, and stance. The next day we had a large poetry reading in the library. In this poetry reading we got to put our names into a ballot box where the poets would call out our names and then we would be able to recite one piece of poetry that we wrote to all our peers, teachers, and our guest poets.

Math: In math we did a lot of logical and debatable reasoning and discussing. So one thing that we did was playing the role of hospital in a country where liver transplants were needed and were very rare to get, so we had to pick out of 10 people which 4 we would save. To decide which people to save we had to think of the important factors such as age, importance, role, money, etc. The next thing we did was model people in the UN who represent different countries. Based on our countries per family monthly income we would debate over food, shelter, and medical care. The country that debated for the most money would win the basic needs debated on, but we soon came to the decision that this was unfair. It was unfair because some countries weren’t as fortunate so they weren’t getting anything, so then we came up with an agreement that suited most of the countries.

PE: With a partner or alone we carried different sizes of buckets around the track for two laps. This was to stimulate how people in poor countries who didn’t have access to clean water nearby had to walk long and tiring journeys just so that they would be able to survive. They needed to take 10 mile walks at the most just to get something that we all take for granted- water. This was a very tiring experience and it made my arm, back, and legs hurt. It was also very hard to walk without spilling any water under the hot sun.

Science: In science we the most interesting game which actually had a lot of meaning hidden behind it. In this game we were split up by sides and needed to get a source of food, water, and medical aid. We had a box that needed to be filled with rice grains to symbolize food, another box that needed to be filled with water to symbolize water, and a box that needed to be filled with 20 ribbon to symbolize medical aid. We had to in pairs had to run into science rooms to get these supplies and the quickest team to get them won. But what we realized was that the game was also unfair. A side was closer than any side to all the facilities, the teachers were helping them, and they had bigger spoons and cups to get water and food with. This conveyed what different countries were going through and how sometimes things can be unjust. In developing countries things are harder to get and sometimes resources are limited, but in more developed countries they have more advantages and they can get their basic needs.

Food Festival: All of the students brought in food that was connected with a country that they felt was important to them. Then they brought it to the cafeteria where we had 3 tables of a variety of food. Then for lunch we got to experience different kinds of food from around the world.

2. Choose two questions from the list below to answer thoroughly, using descriptions and detail.

• Did you learn a new skill or discover an interest during UN Day?

A new interest that I have discovered was debating with reasoning and learning about how the UN works. We participated in this activity in Math class and I really enjoyed it. I loved how we got to debate with other students in the class over who to give a liver transplant to so that they could survive. I like understanding about the individuals could and have affected the country and their contributions to the development of the country. I also liked the idea of being a country in the UN and having a certain amount of money and coming up with solutions to fix the unjust. This stimulation was very real because we even learnt that the countries who have seemed to won world war II had more power over voting and the decisions that were made.

• During UN Day, did you hear, smell, or feel anything that surprised you?

I felt a lot of unjust, unfairness, and sorrow during UN Day. When we were stimulating the game in science of getting our basic needs I felt as if we were being cheated. As if something wasn’t right and some countries (sides) were being treated differently because of different circumstances. I also felt really fortunate and sad for the people who carry these huge buckets of water for long distances underneath the hot sun. This taught me not to take water and my life for granted. Also in math I felt kind of bad for the groups who were poor countries and didn’t have a lot of money. They really didn’t have any choice and they couldn’t do anything about their situation. This got me to thinking about how the UN has a very important purpose to make things as fair as possible.

• How was your UN Day experience different from what you expected?

I thought that the UN day was like what we did in elementary school. I thought that we would learn about all the different cultures in the world and do activities based on those cultures. But actually this experience was actually much funner and more engaging. It taught us something different about the world and gave us a new perspective on life. This was a very engaging day as it was something new. Most of us who have been at SAS for a long time are used to the normal UN day but this was different and funner. I also really liked how the activities that we did were not only entertaining but also taught us a lot through them.  

• What do you view differently now because of your experiences during UN Day?

I view the difference between developing, almost developed, and developed countries differently and how important the UN is. I already knew that there was a difference between the amount of money the countries made and the difference between the qualities of life. I already know that the UN helps countries with their issues and solves problems. But now I realize that the countries maybe just needed some help, maybe they felt as if they weren’t being treated fairly. Maybe they just needed that little boost that advantage. If that wasn’t possible then maybe they just needed an agreement something that sets them the same as other countries something that makes them equal. And that’s why they have the UN, to solve problems, come up with solutions, make things fair and right.

3. Choose 1 question from the list below to answer thoroughly, using descriptions and detail.

• If you could do UN Day again, what would you do the same or differently?

If I were to change anything about UN day I would change the RLA and PE activities. I’m not sure if the RLA activity was because of the authors or if it had to do with UN day for a reason. Maybe we could have a more related activity which helps teach us about the UN and the countries around the world. Maybe the activity was connected through public speaking in the UN, I’m not sure, so maybe they could have explained that. Also in PE some people were spilling 3/4 of the water out of their buckets so that the bucket would be lighter and easier for them to run around the track with twice. I thought that this was very bad and it just defeated the whole purpose of doing the activity. If the people weren’t giving it their all then why even bother, they wouldn’t have really understood what the less fortunate people were going through.

Looking for Pythagoras- Math Reflection Questions Investigation 4

1. Give three examples of fractions with decimal representations that terminate:
A terminating decimal is a decimal that has digits that doesn't go on forever. A terminating decimal is the same thing as a rational number, and doesn't repeat forever. Three examples of fractions with decimal representations that terminate are 1/8,  2/10,  and 1/4. The decimal representation of 1/8 is equal to 0.125, which is terminating because the decimal values don't go on forever. The decimal representation of 2/10 is equal 0.2, which is terminating because the decimal values don't go on forever. The decimal representation of 1/4 is equal to 0.25, which is terminating because the decimal values don't go on forever. 

2. Give three examples of fractions with decimal representations that repeat:
A repeating decimal is when the decimal values of the number goes on forever, or infinitely. Three examples of fractions with decimal representations that repeat are 1/3, 1/11, and 2/9. The decimal representation of 1/3 is 0.3333333...... (3 repeating on forever). This is an example of a repeating decimal because the three goes on forever. The decimal representation of 1/11 is 0.090909...... (09 repeating on forever). This is an example of a repeating decimal because the three goes on forever. The decimal representation of 2/9 is 0.222222...... (2 repeating on forever). This is an example of a repeating decimal because the three goes on forever. Usually, repeating decimals are shown with a bar over the number or numbers that are getting repeated.

3. Give three examples of irrational numbers, including one irrational number greater than 5:
An irrational number is a number that cannot be expressed as a ratio or a fraction. Three examples of irrational numbers, including one that is greater than 5, are √27, √30, and √34. The square root of 27 is approximately (fraction estimate) 5.196. The square root of 30 is approximately 5.477. The square root of 34 is approximately 5.8309. But remember that these square roots have no ends and the answers that I gave were just estimates. 

4. How can you determinate whether you can write a given decimal as a fraction:
You can determine whether you can write a given decimal as a fraction by looking at it. You automatically know that a decimal can be represented as a fraction if it is either terminating or rational. You also know that if something irrational, that it CAN'T be written as a fraction. If a fraction neither repeats nor terminates, you automatically know that it can't be written as a fraction, which means that it is irrational. When looking at the terminating decimal, you can look at the place value in which the last number is in. For example, let's use 0.02. THe 2 is in the hundredths place, so the fraction is 2/100. You could also guess and check to make the decimal a fraction.