Contents No. | Title | Page | 1 | Introduction | 2 | 2 | Part 1 | 6 | 3 | Part 2 | 8 | 4 | Part 3 | 10 | 5 | Part 4 | 13 | 6 | Part 5 | 17 | 7 | Further Exploration | 21 | 8 | Reflection | 22 |
Introduction
Moral Values: I have learned many moral values while completing this assignment. Better still, I got to know about the importance of applying these moral values into our daily lives. The first moral value is to cooperate with other people. The strategies and solutions of the questions in this project work were discussed among me and a group of friends. This makes things easier and saved a lot of time. The management of time is also important to complete this project work. Other than this assignment, I have homework, extra co curricular activities and tuition classes to attend. Thus a good management of time is essential for me to complete this given task and not to disrupt my daily activities. Perseverance has taught me to be steady and persistent in doing something. In spite of many difficulties I faced throughout the whole procedure I learned that giving up is just not right solution. Obstacles and discouragement should be endured the course of action should be held on with unyielding determination to see obtain sweet fruit of success. I had also learned to appreciate the beauty of mathematics. Waxing eloquently on the basic importance of Mathematics in human life, Roger Bacon (1214-1294), an English Franciscan friar, philosopher, scientist and scholar of the 13th century, once stated: "Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world." And the ingenuity of his statement is there before us to see, in this Internet era.
Objectives:
My objectives to achieve upon completion of the Additional Mathematics Project Work are:
1. to realize the importance and the beauty of mathematics.
2. to prepare myself for the demands of my future undertakings and in workplace.
3. to use technology especially the ICT appropriately and effectively.
4. to acquire effective mathematical communication through writing, and to use the language of mathematics to express mathematical ideas correctly and precisely.
5. to apply and adapt a variety of problem-solving strategies to solve routine and non-routine problems.
Introduction to probabilities:
Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such area of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and underlying mechanics of complex systems. The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later. history of probabilities:
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Etymology
Probable and likely and their cognates in other modern languages derive from medieval learned Latin probabilis and verisimilis, deriving from Cicero and generally applied to an opinion to mean plausible or generally approved.
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Origins
Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptions and half-proof to deal with the uncertainties of evidence in court. In Renaissance times, betting was discussed in terms of odds such as "ten to one" and maritime insurance premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums.
The mathematical methods of probability arose in the correspondence of Pierre de Fermat and Blaise Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance. Christiaan Huygens (1657) gave a comprehensive treatment of the subject.
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18th Century
Jacob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's The Doctrine of Chances (1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large numbers, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).
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Nineteenth century
The power of probabilistic methods in dealing with uncertainty was shown by Gauss's determination of the orbit of Ceres from a few observations. The theory of errors used the method of least squares to correct error-prone observations, especially in astronomy, based on the assumption of a normal distribution of errors to determine the most likely true value.
Towards the end of the nineteenth century, a major success of explanation in terms of probabilities was the Statistical mechanics of Ludwig Boltzmannand J. Willard Gibbs which explained properties of gases such as temperature in terms of the random motions of large numbers of particles.
The field of the history of probability itself was established by Isaac Todhunter's monumental History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865).
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Twentieth century
Probability and statistics became closely connected through the work on hypothesis testing of R. A. Fisher and Jerzy Neyman, which is now widely applied in biological and psychological experiments and in clinical trials of drugs. A hypothesis, for example that a drug is usually effective, gives rise to a probability distribution that would be observed if the hypothesis is true. If observations approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected.
The theory of stochastic processes broadened into such areas as Markov processes and Brownian motion, the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets, leading to the use of sophisticated probability models in mathematical finance, including such successes as the widely-used Black-Scholes formula for the valuation of options.
The twentieth century also saw long-running disputes on the interpretations of probability. In the mid-century frequentism was dominant, holding that probability means long-run relative frequency in a large number of trials. At the end of the century there was some revival of the Bayesian view, according to which the fundamental notion of probability is how well a proposition is supported by the evidence for it.
The mathematical treatment of probabilities, especially when there are infinitely many possible outcomes, was facilitated by Kolmogorov's axioms(1931). examples of the probability theory applications:
Weather forecasting
Suppose you want to go on a picnic this afternoon, and the weather report says that the chance of rain is 70%? Do you ever wonder where that 70% came from?
Forecasts like these can be calculated by the people who work for the National Weather Service when they look at all other days in their historical database that have the same weather characteristics (temperature, pressure, humidity, etc.) and determine that on 70% of similar days in the past, it rained.
As we've seen, to find basic probability we divide the number of favorable outcomes by the total number of possible outcomes in our sample space. If we're looking for the chance it will rain, this will be the number of days in our database that it rained divided by the total number of similar days in our database. If our meteorologist has data for 100 days with similar weather conditions (the sample space and therefore the denominator of our fraction), and on 70 of these days it rained (a favorable outcome), the probability of rain on the next similar day is 70/100 or 70%.
Since a 50% probability means that an event is as likely to occur as not, 70%, which is greater than 50%, means that it is more likely to rain than not. But what is the probability that it won't rain? Remember that because the favorable outcomes represent all the possible ways that an event can occur, the sum of the various probabilities must equal 1 or 100%, so 100% - 70% = 30%, and the probability that it won't rain is 30%.
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Batting averages
Let's say your favorite baseball player is batting 300. What does this mean?
A batting average involves calculating the probability of a player's getting a hit. The sample space is the total number of at-bats a player has had, not including walks. A hit is a favorable outcome. Thus if in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For baseball stats we multiply all the percentages by 10, so a 30% probability translates to a 300 batting average.
This means that when a Major Leaguer with a batting average of 300 steps up to the plate, he has only a 30% chance of getting a hit - and since most batters hit below 300, you can see how hard it is to get a hit in the Major Leagues!
Part 1
Question:
The theory of probability has been applied in various fields such as market research, medical research, transportation, business, management and so on.
(a) Conduct research on the history of probability and give at least two examples on how the theory of probability is being applied in real life situations. Then, write an Introduction to this Project Work based on your findings. You may include the historical aspects, examples of the probability theory applications and its importance to real life situations.
(b) The probability theory can be divided into two categories: Theoretical Probabilities and Empirical Probabilities. Find out, discuss and write about the difference between the Theoretical and Empirical Probabilities. Problem statement:
1. What is the history of probability and how is the theory being applied into our daily lives?
2. What is the difference between the Theoretical and Empirical Probabilities? Strategy:
1. Search for information by surfing the Internet for the history of probability and how is the theory being applied into our daily lives and write and Introduction to this Project Work based on the findings.
2. The definition of Theoretical and Empirical Probabilities were looked at from the reference book in the school library.
3. The comparison of Theoretical and Empirical Probabilities were shown by using a solution to highlight the difference.
Solution:
B) Difference between the Theoretical and Empirical Probabilities
The term empirical means "based on observation or experiment." An empirical probability is generally, but not always, given with a number indicating the possible percent error (e.g. 80+/-3%). A theoretical probability, however, is one that is calculated based on theory, i.e., without running any experiments.
Empirical Probability of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences.
Theoretical Probability of an event is the number of ways that the event can occur, divided by the total number of outcomes. It is finding the probability of events that come from a sample space of known equally likely outcomes.
Comparing Empirical and Theoretical Probabilities: Karen and Jason roll two dice 50 times and record their results in the accompanying chart.
1.) What is their empirical probability of rolling a 7?
2.) What is the theoretical probability of rolling a 7?
3.) How do the empirical and theoretical probabilities compare? | Sum of the rolls of two dice | 3, 5, 5, 4, 6, 7, 7, 5, 9, 10,
12, 9, 6, 5, 7, 8, 7, 4, 11, 6,
8, 8, 10, 6, 7, 4, 4, 5, 7, 9,
9, 7, 8, 11, 6, 5, 4, 7, 7, 4,
3, 6, 7, 7, 7, 8, 6, 7, 8, 9 | | Solution:
1.) Empirical probability (experimental probability or observed probability) is 13/50 = 26%.
2.) Theoretical probability (based upon what is possible when working with two dice) = 6/36 = 1/6 = 16.7% (check out the table at the right of possible sums when rolling two dice).
3.) Karen and Jason rolled more 7's than would be expected theoretically. | | conclusion: Empirical and theoretical probabilities are both different ways to calculate the probability. The results of an experiment may manipulate the answer of empirical probability but theoretical probability will not be affected and may not even need to conduct an experiment.

Part 2
Question:
(a) Suppose you are playing the Monopoly game with two of your friends. To start the game, each player will have to toss the die once. The player who obtains the highest number will start the game. List all the possible outcomes when the die is tossed once. (b) Instead of one die, two dice can also be tossed simultaneously by each player. The player will move the token according to the sum of all dots on both turned-up faces. For example, if the two dice are tossed simultaneously and "2" appears on one die and "3" appears on the other, the outcome of the toss is (2, 3). Hence, the player shall move the token 5 spaces. Note: The events (2, 3) and (3, 2) should be treated as two different events. List all the possible outcomes when two dice are tossed simultaneously. Organize and present your list clearly. Consider the use of table, chart or even tree diagram.
Problem Statement:
1. What is the possible outcome when the die is tossed once?
2. What are the possible outcomes when 2 dices are tossed simultaneously?
Strategy:
1. The possible outcome when the die is tossed once each is figured out by knowing that all numbers on the die is possible to be the outcome.
2. The possible outcomes when 2 dices are tossed simultaneously is figured out by using a chart to list all the outcomes.
Solution:
a) {1, 2, 3, 4, 5, 6}

b) Chart Dice 2

6 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) 5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) 4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) 3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) 2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) 1 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) Dice 1 0 1 2 3 4 5 6

conclusion:
There are 6 possible outcomes for (a) which are {1, 2, 3, 4, 5, 6}. While there are 36 possible outcomes of (b) which are (1,6), (2,6), (3,6), (4,6), (5,6), (6,6), (1,5), (2,5), (3,5), (4,5), (5,5), (6,5), (1,4), (2,4), (3,4), (4,4), (5,4), (6,4), (1,3), (2,3), (3,3), (4,3), (5,3), (6,3), (1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (1,1), (2,1), (3,1), (4,1), (5,1) and (6,1).

Part 3
Question:
Table 1 shows the sum of all dots on both turned-up faces when two dice are tossed simultaneously.
(a) Complete Table 1 by listing all possible outcomes and their corresponding probabilities.

(b) Based on Table 1 that you have completed, list all the possible outcomes of the following events and hence find their corresponding probabilities:
A = {The two numbers are not the same)
B = {The product of the two numbers is greater than 36}
C = {Both numbers are prime or the difference between two numbers is odd) I) = {The sum of the two numbers are even and both numbers are prime)

problem statement:
1. What are the possible outcomes and their corresponding probabilities for the sum of all dots on both turned-up faces when two dice are tossed simultaneously?
2. What are the possible outcomes of the following events and their corresponding probabilities?
Strategy:
1. The possible outcomes are all listed in table form.
2. The probability is calculated by dividing the number of outcome with 36.
Solution:
a) Table 1 show the sum of all dots on both turned-up faces when two dice are tossed simultaneously. Sum of the dots on both turned-up faces (x) | Possible outcomes | Probability, P(x) | 2 | (1,1) | 1/36 | 3 | (1,2),(2,1) | 2/36 | 4 | (1,3),(2,2),(3,1) | 3/36 | 5 | (1,4),(2,3),(3,2),(4,1) | 4/36 | 6 | (1,5),(2,4),(3,3),(4,2),(5,1) | 5/36 | 7 | (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) | 6/36 | 8 | (2,6),(3,5),(4,4),(5,3),(6,2) | 5/36 | 9 | (3,6),(4,5),(5,4),(6,3) | 4/36 | 10 | (4,6),(5,5),(6,4) | 3/36 | 11 | (5,6),(6,5) | 2/36 | 12 | (6,6) | 1/36 |

b) Table of possible outcomes of the following events and their corresponding probabilities. Events | Possible outcomes | Probability,P(x) | A | {(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5) } | 3036 | B | ø | ø | C | P = Both number are primeP = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}Q = Difference of 2 number is oddQ = { (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4),(3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5) }C = P U QC = {1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,5), (3,2), (3,3), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,5) } | 2236 | D | P = Both number are primeP = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}R = The sum of 2 numbers are evenR = {(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2(, (6,4), (6,6)}D = P ∩ RD = {(2,2), (3,3), (3,5), (5,3), (5,5)} | 536 | part 4 question: (a) Conduct an activity by tossing two dice simultaneously 50 times. Observe the sum of all dots on both turned-up faces. Complete the frequency table below.

Based on Table 2 that you have completed, determine the value of:
(i) mean;
(ii) variance; and
(iii) standard deviation of the data.
(b) Predict the value of the mean if the number of tosses is increased to 100 times.
(c) Test your prediction in (b) by continuing Activity 3(a) until the total number of tosses is 100 times. Then, determine the value of:
(i) mean;
(ii) variance; and
(iii) standard deviation of the new data.
Was your prediction proven? problem statement:
1. What is the sum of all dots on both turned-up faces when two dice were tossed simultaneously for 50 times?
2. How to predict the value of the mean if the number of tosses is increased to 100 times?
3. How to prove the prediction?
Strategy:
1. A table is drawn to list down the sum of the two numbers (x), frequency (f), fx and fx2.
2. The mean, variance and standard deviation were calculated using mathematical skills learned in school.
3. Another table is drawn to list down the sum of the two numbers (x), frequency (f), fx and fx2 for 1000 tosses.
3. The mean, variance and standard deviation were calculated for the second table to prove the prediction. solution: a) Sum of the two numbers (x) | Frequency (f) | fx | fx2 | 2 | 2 | 4 | 8 | 3 | 4 | 12 | 36 | 4 | 4 | 16 | 64 | 5 | 9 | 45 | 225 | 6 | 4 | 24 | 144 | 7 | 11 | 77 | 539 | 8 | 4 | 32 | 256 | 9 | 6 | 54 | 486 | 10 | 3 | 30 | 300 | 11 | 1 | 11 | 121 | 12 | 2 | 24 | 288 | f = 50 | fx = 329 | fx2= 2467 |
Table 2

i) Mean = x = fxf = 32950 = 6.58 ii) Variance = = fx2f - x 2 = 246750 – (6.58)2 = 6.044 iii) Standard deviation = ( fx2f - x 2 ) = 6.0436
= 2.458
b)
Sum of the two numbers (x) | Frequency (f) | fx | fx2 | 2 | 4 | 8 | 16 | 3 | 5 | 15 | 45 | 4 | 6 | 24 | 96 | 5 | 16 | 80 | 400 | 6 | 12 | 72 | 432 | 7 | 21 | 147 | 1029 | 8 | 10 | 80 | 640 | 9 | 8 | 72 | 648 | 10 | 9 | 90 | 900 | 11 | 5 | 55 | 605 | 12 | 4 | 48 | 576 | f = 100 | fx = 691 | fx2= 5387 |

Prediction of mean = 6.91 i. Mean = 691100 = 6.91 ii. Variance = fx2f - x 2 = 5387100- (6.91)2 = 6.122 iii. Standard deviation = 6.122 = 2.474 Prediction is proven.

Part 5 question: When two dice are tossed simultaneously, the actual mean and variance of the sum of all dots on the turned-up faces can be determined by using the formulae below:

(a) Based on Table 1, determine the actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces by using the formulae given.

(b) Compare the mean, variance and standard deviation obtained in Part 4 and Part 5. What can you say about the values? Explain in your own words your interpretation and your understanding of the values that you have obtained and relate your answers to the Theoretical and Empirical Probabilities.
(c) If n is the number of times two dice are tossed simultaneously, what is the range of mean of the sum of all dots on the turned-up faces as n changes? Make your conjecture and support your conjecture. problem statement:
1. What is the actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces?
2. What are the comparisons between the mean, variance and standard deviation obtained in Part 4 and Part 5? strategy: 1. The actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces using mathematical methods.
2. The mean, variance and standard deviation obtained in Part 4 and Part 5 were tabulated and compared. solution: a)
Mean = x P(x)
= 2136+3118+4112+519+6536+716+8536+919+10112+11118+ 12136
= 7
Variance = xP(x) – (mean)
= 22136+32118+42112+5219+62536+7216+82536+9219+102112+ 112118+122136 - (7)2
= 54.83 – 49
= 5.83
Standard deviation = 5.83 = 2.415

b) | Part 4 | Part 5 | | n = 50 | n = 100 | | Mean | 6.58 | 6.91 | 7.00 | Variance | 6.044 | 6.122 | 5.83 | Standard deviation | 2.458 | 2.474 | 2.415 |

We can see that, the mean, variance and standard deviation that we obtained through experiment in part 4 are different but close to the theoretical value in part 5.

For mean, when the number of trial increased from n=50 to n=100, its value get closer (from 6.58 to 6.91) to the theoretical value. This is in accordance to the Law of Large Number. We will discuss Law of Large Number in next section.

Nevertheless, the empirical variance and empirical standard deviation that we obtained i part 4 get further from the theoretical value in part 5. This violates the Law of Large Number. This is probably due to a. The sample (n=100) is not large enough to see the change of value of mean, variance and standard deviation. b. Law of Large Number is not an absolute law. Violation of this law is still possible though the probability is relative low. conclusion: The empirical mean, variance and standard deviation can be different from the theoretical value. When the number of trial (number of sample) getting bigger, the empirical value should get closer to the theoretical value. However, violation of this rule is still possible, especially when the number of trial (or sample) is not large enough.

c)
The range of the mean

Conjecture: As the number of toss, n, increases, the mean will get closer to 7. 7 is the theoretical mean.

Image below support this conjecture where we can see that, after 500 toss, the theoretical mean become very close to the theoretical mean, which is 3.5. (Take note that this is experiment of tossing 1 die, but not 2 dice as what we do in our experiment)

FURTHER EXPLORATION In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
For example, a single roll of a six-sided die produces one of the numbers 1, 2, 3, 4, 5, 6, each with equal probability. Therefore, the expected value of a single die roll is

According to the law of large numbers, if a large number of dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the accuracy increasing as more dice are rolled. Similarly, when a fair coin is flipped once, the expected value of the number of heads is equal to one half. Therefore, according to the law of large numbers, the proportion of heads in a large number of coin flips should be roughly one half. In particular, the proportion of heads after n flips will almost surely converge to one half as approaches infinity. Though the proportion of heads (and tails) approaches half, almost surely the absolute (nominal) difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as number of flips becomes large. Also, almost surely the ratio of the absolute difference to number of flips will approach zero. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows. The LLN is important because it "guarantees" stable long-term results for random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will converge to the expected value or that a streak of one value will immediately be "balanced" by the others.
In conclusion, the law of large numbers is more frequently used in casinos or calculations than empirical and theoretical probabilities due to its long-term stability and accuracy.
REFLECTION
While I was conducting the project, I learned the moral values that I practiced to complete this Project Work. This Project Work had taught me to be more confident when doing something especially the homework given by the teacher. I also learned to be a disciplined type of student which is always sharp on time while doing some work, complete the work by myself and researching the information from the internet.
With the information obtained while completing this assignment, I get to know the usefulness and importance of the possibility theory in calculations. Many ways to calculate probability were also learned via research using the Internet.

The End!