Add Math Project 2 - Copy

8/8/2019 Add Math Project 2 - Copy

1/23

1

Introduction

Probability theory is the branch of mathematics concerned with analysis of random

phenomena. The central objects of probability theory are random variables, stochastic

processes, and events: mathematical abstractions of non-deterministic events or measured

quantities that may either be single occurrences or evolve over time in an apparently random

fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many

times the sequence of random events will exhibit certain statistical patterns, which can be

studied and predicted. Two representative mathematical results describing such patterns are

the law of large numbers and the central limit theorem. As a mathematical foundation for

statistics, probability theory is essential to many human activities that involve quantitative

analysis of large sets of data. Methods of probability theory also apply to descriptions of

complex systems given only partial knowledge of their state, as in statistical mechanics. A great

discovery of twentieth century physics was the probabilistic nature of physical phenomena at

atomic scales, described in quantum mechanics.


2/23

2

Objective

The aims of carrying out this project work are :

i. to apply and adapt a variety of problem-solving strategies to solve

problem;

ii. to improve thinking skills;

iii. to promote effective mathematical communication;

iv. to develop mathematical knowledge through problem solving in a way that increases

students interest and confidence;

v. to use the language of mathematics to express mathematical ideas precisely;

vi. to provide learning environment that stimulates and enhances effective learning;

vii. to develop positive attitude towards mathematics.

Moral Values

While I was conducting this project, I had learnt many moral values that I practice. This

project work had taught me to be more confident when doing something especially the

homework given by my teacher. I also learnt to be disciplined and not procrastinate while doing

a task. I had learnt to be independent to complete the work by myself through researching the

information from the internet.


3/23

3

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.

Answer

Part 1 (a)

History

A gambler's dispute in 1654 led to the creation of a mathematical theory of probability by two

famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud,

Chevalier de Mr, a French nobleman with an interest in gaming and gambling questions,

called Pascal's attention to an apparent contradiction concerning a popular dice game. The

game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not

to bet even money on the occurrence of at least one "double six" during the 24 throws. A

seemingly well-established gambling rule led de Mr to believe that betting on a double six in

24 throws would be profitable, but his own calculations indicated just the opposite.

This problem and others posed by de Mr led to an exchange of letters between Pascal

and Fermat in which the fundamental principles of probability theory were formulated for the

first time. Although a few special problems on games of chance had been solved by some Italianmathematicians in the 15th and 16th centuries, no general theory was developed before this

famous correspondence.

The Dutch scientist Christian Huygens, a teacher of Leibniz, learned of this correspondence

and shortly thereafter (in 1657) published the first book on probability; entitled De Ratiociniis in

Ludo Aleae, it was a treatise on problems associated with gambling. Because of the inherent


4/23

4

appeal of games of chance, probability theory soon became popular, and the subject developed

rapidly during the 18th century. The major contributors during this period were Jakob Bernoulli

(1654-1705) and Abraham de Moivre (1667-1754).

In 1812 Pierre de Laplace (1749-1827) introduced a host of new ideas and mathematical

techniques in his book, Thorie Analytique des Probabilits. Before Laplace, probability theory

was solely concerned with developing a mathematical analysis of games of chance. Laplace

applied probabilistic ideas to many scientific and practical problems. The theory of errors,

actuarial mathematics, and statistical mechanics are examples of some of the important

applications of probability theory developed in the l9th century.

Like so many other branches of mathematics, the development of probability theory has

been stimulated by the variety of its applications in different fields as genetics, psychology,

economics, and engineering. Many workers have contributed to the theory since Laplace's

time; among the most important are Chebyshev, Markov, von Mises, and Kolmogorov.

In 1933 a monograph by a Russian mathematician A. Kolmogorov outlined an axiomatic

approach that forms the basis for the modern theory. (Kolmogorov's monograph is available in

English translation as Foundations of ProbabilityTheory, Chelsea, New York, 1950). Since then

the ideas have been refined somewhat and probability theory is now part of a more general

discipline known as measure theory.

PROBABILITY THEORY

Like other theories, the theory of probability is a representation of probabilistic

concepts in formal terms that is, in terms that can be considered separately from their

meaning. These formal terms are manipulated by the rules of mathematics and logic, and

any results are then interpreted or translated back into the problem domain. There has been at

least two successful attempts to formalize probability, namely the Kolmogorov formulation and

the Cox formulation. In Kolmogorov's formulation (probability space),sets are interpreted as

events and probability itself as a measure on a class of sets. In Cox's theorem, probability is

taken as a primitive and not further analyzed. The emphasis is based on constructing a

consistent assignment of probability values to propositions. In both cases, the laws of

probability are the same, except for technical details.


5/23

5

Applications of probability in our daily lives

(i) 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 weatherconditions (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%.

(ii) Batting averages

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, it is

demonstrated that it is hard to get a consistent hit in the Major Leagues.


6/23

6

Part 1 (b)

Probability is a likelihood that an event will happen.

Empirical probability is where the probability has been determined by repeated attempts.The

term empirical means "based on observation or experiment". Dangerous medical procedurescan also have empirical probability playing as a factor. There is always a chance that someone

dies under the knife, or that someone cures on their own. Based on those odds, a doctor could

advise for or against certain procedures. Those odds are based on other patients who have

gone through the same thing.

Theoretical probability is where the probability has been calculated using predetermined

variables used as a model

We can find the theoretical probability of an event using the following ratio:

Gambling is about theoretical probability. One can assume that all the chips, cards, tables or

whatever are completely fair (or even calculate the unfairness, based on the method of

shuffling), so one can calculate the odds of a certain set of cards coming up, before they ever

have.

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


7/23

7

Solution:

1) Empirical probability (experimental probability or observed

probability) is 13/50 = 26%.

2) Theoretical probability (based upon what is possible whenworking 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.

Differencebetween the Theoretical and Empirical Probabilities

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 samplespace of known equally likely outcomes.


8/23

8

Part 2

(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.

Part 2(a)

When we play monopoly, we have to toss the die once to find who is going to start the game

first. The possible outcomes when we toss the die is 1,2,3,4,5, and 6. This is because a die has 6

surface as shown in the figure below.

P(Outcomes) = {1, 2, 3, 4, 5, 6}


9/23

9

Part 2(b)

Table

1 2 3 4 5 6

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

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

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

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

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

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

Chart


10/23

10

Tree diagram


11/23

11

Part 3

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)


12/23

12

Part 3 (a)

Table 1

Sum of the dots on

both turned up faces

Possible outcomes Probability

P(X)

2 (1,1) 1/36

3 (1,2), (2,1) 2/36 = 1/18

4 (1,3), (2,2), (3,1) 3/36 = 1/12

5 (1,4), (2,3), (3,2), (4,1) 4/36 = 1/9

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 = 1/6

8 (2,6), (3,5), (4,4), (5,3), (6,2) 5/36

9 (3,6), (4,5), (5,4), (6,3) 4/36 = 1/9

10 (4,6), (5,5), (6,4) 3/36 = 1/12

11 (5,6), (6,5) 2/36 = 1/18

12 (6,6) 1/36


13/23

13

Part 3 (b)

Based on the table,

A = {The two numbers are not the same}

A = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3),

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

(6,2), (6,3), (6,4), (6,5), (6,6)}

B = {The product of the two numbers is greater then 36}

B = {Does not exist}

B =

Let P = {Both prime numbers}

P = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}

Let Q = {Difference between 2 numbers is odd}

Q = { (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 Q

C= {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) }

Let R = {The sum of 2 numbers are even}

R = {(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)}


14/23

14

Part 4

(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:

(a)(i) mean;

(ii) variance; and

(iii) standard deviationof 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 data.Was your prediction proven?


15/23


16/23

16

Part 4 (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 90011 5 55 605

12 4 48 576

= 100 = 691 = 5387

Prediction of mean = 6.91

Mean =

= 6.91

Variance = -

= (6.91)

2

= 6.122

Standard deviation = = = 2.474

Prediction is proven.


17/23

17

Part 5

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


18/23

18

Part 5 (a)

(i) Mean = xP(x)

=

= 7

(ii) Variance = x2

P(x) (mean)2

= - (7)2

= 54.83 49

= 5.83

(iii) Standard deviation = = 2.415


19/23

19

Part 5 (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

The mean, variance and standard deviation obtained through the experiment in part 4 are

different because Part 4 was the Empirical Probability and Part 5 was the Theoretical

Probability. For example, the empirical probability of rolling a 7 is 4/25 = 16%. But the

theoretical probability of rolling a 7 is 6/36 = 1/6 = 16.7%. The table above shows thecomparison between Mean, Variance, and Standard Deviation.

For the mean, its value get closer (from 6.58 to 6.91) to the theoretical value when the number

of trial increased from n=50 to n=100. This is in accordance to the Law of Large Number.

Nevertheless, the empirical variance and empirical standard deviation that obtained in part 4

becomes 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.

In 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.


20/23

20

PART 5 (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.

The graph below supports this conjecture. However, this is experiment of tossing 1 die, not 2

dice as what we do in our experiment. As presented, after 500 tosses, the theoretical mean

become very close to the theoretical mean, which is 3.5.

Conclusion

The answer for Part 4 is different from Part 5 although the questions are similar. I can conclude

that the mean, variance, and standard deviation changes although the number of tossed of the

dice was increased until 100 times because when the number of tossed changes, the frequency

also changes.


21/23

21

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 tothe 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 equalprobability. 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 napproaches 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 numberof

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.


22/23

22

Conclusion

From the experience I have gained from the completion of this project, I have concluded

that probabilities of events depend on whether the events are independent events or not.

In conclusion, the empirical mean, variance and standard deviation can be different from

the theoretical value. When the number of trials of an event increases, the empirical value gets

closer to the theoretical value. Anyhow, an exception should be made especially when the

number of trials is few. I have constructed a table below to show comparison between

Questions 4 and 5 despite the similarity found in the two questions.

Part 4 Part 5

n = 50 n = 100

Mean 6.58 6.91 7.00

Variance 6.044 6.122 5.83Standard deviation 2.458 2.474 2.415

From drawing the table above, I have taken note that for different events although the

questions are similar, the answer might be different. I can conclude that the mean, variance,

and standard deviation changes if the frequency also changes. I have confirmed that the values

of the mean, variance and standard deviation changes with frequency.


23/23

23

Reflection

Through the completion of this project work for Additonal Mathematics 2010, I have studied

the method of evaluating probabilities of events relating to independent events and conditionalevents. I have learnt that an experiment repeated under essentially homogeneous and similar

conditions results in an outcome, which is unique or not unique but may be one of the several

possible outcomes. I understand now that when the result is unique then the experiment is

called a deterministic experiment.

This project had also taught me to responsible in completing the tasks given to me. Attempting

to finish this project had trained me not to give up easily when I could not find the solution for

the question. Actually, I also enjoyed doing this project during my school holiday as I was able

to spend my time with friends to complete this project together and it had tightened ourfriendship.

Add Math Project 2 - Copy

Documents

Transcript of Add Math Project 2 - Copy