Additional Mathematics Project2010

Assignment 2

Name: Nur Syafiqah Amani Bt AzmiForm: 5 Beta

I/C Number: 931224-03-5700Teacher’s Name: Pn. Sharifah Nur

Afizah

From Calculus, Volume II by Tom M. Apostol (2nd edition, John Wiley & Sons, 1969):

"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 Méré, 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 Méré 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 Méré 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 Italian mathematicians 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 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, Théorie Analytique des Probabilités. 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. Conversely, each advance in the theory has enlarged the scope of its influence. Mathematical statistics is one important branch of applied probability; other applications occur in such widely 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.

One of the difficulties in developing a mathematical theory of probability has been to arrive at a definition of probability that is precise enough for use in mathematics, yet comprehensive enough to be applicable to a wide range of

phenomena. The search for a widely acceptable definition took nearly three centuries and was marked by much controversy. The matter was finally resolved in the 20th century by treating probability theory on an axiomatic basis. 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 Probability Theory, 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 in our 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 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%.

ii) 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!

Probability is a way of expressing knowledge or belief that an event will occur or has occurred.

In mathematicsthe concept has been given an exact meaning

in probability theory, that is used extensively in such areas of study as

mathematics, statistics, finance, gambling, science, and philosophy to draw

conclusions about the likelihood of potential events and the underlying

mechanics of complex systems. For example, we widely used probability in gambling and

weather forecasting.

http://www.cc.gatech.edu/classes/ cs6751_97_winter/Topics/stat-meas/probHist.html

http://www.regentsprep.org/Regents/math/ ALGEBRA/APR5/theoProp.htm

EMPIRICAL PROBABILITY

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.

Empirical Probability Formula

P(E) = probability that an event, E, will occur.top = number of ways the specific event occurs.bottom =  number of ways the experiment could

occur.

 

Example:   A survey was conducted to determine students' favourite breeds of dogs.  Each student chose only one breed. 

Dog Collie Spaniel Lab Boxer

PitBull Other

# 10 15 35 8 5 12

What is the probability that a student's favourite dog breed is Lab?Answer:  35 out of the 85 students chose Lab.  The

probability is

THEORICAL PROBABILITY

Theorical 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.

Theoretical Probability Formula

P(E) = probability that an event, E, will occur.n(E) = number of equally likely outcomes of E.n(S) = number of equally likely outcomes of sample space S. 

Example 1:   Find the probability of rolling a six on a fair die.

Answer:  The sample space for rolling is die is 6 equally likely results: {1, 2, 3, 4, 5, 6}.

The probability of rolling a 6 is one out of 6 or .

Example 2:   Find the probability of tossing a fair die and getting an odd number.

Answer:event E :  tossing an odd numberoutcomes in E:  {1, 3, 5}sample space S:   {1, 2, 3, 4, 5, 6}

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.

The aims carrying out this project work are:

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

ii. To improve thinking skills;

iii. To promote effective mathematical communication;

iv. To develop mathematical knowledge through problem solving in a way that increases student interest and confident

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.

a) List all the possible outcomes when the dice is tossed once.

There are three player, considered as P1,P2, and P3. The total side of the die which is cube is six,

and the number of dots on the dice is 1, 2, 3, 4, 5 and 6 respectively.

die player 1 2 3 4 5 6

P1 1,1 1,2 1,3 1,4 1,5 1,6P2 2,1 2,2 2,3 2,4 2,5 2,6P3 3,1 3,2 3,3 3,4 3,5 3,6

Therefore, the possible outcome is {1,2,3,4,5,6}

b) List all the possible outcome when two dice are tossed simultaneously.

Possible pair of two dice thrown simultaneously by each plays

dice 1 2 3 4 5 61 1,1 1,2 1,3 1,4 1,5 1,62 2,1 2,2 2,3 2,4 2,5 2,63 3,1 3,2 3,3 3,4 3,5 3,64 4,1 4,2 4,3 4,4 4,5 4,65 5,1 5,2 5,3 5,4 5,5 5,66 6,1 6,2 6,3 6,4 6,5 6,6

n(s)=36

a) Complete table 1 by listing all possible outcomes and their corresponding probabilities.

Sum of the Possible outcomes Probability,

dots on both turned-up faces (x)

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

Table 1

b) Table of all the possible outcomes of the 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) }

B ø ø

CP = 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) }

DP = 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)}

a) Table of an activity by tossing two dice simultaneously 50 times.

Sum of the two numbers ( )

Frequency ( )

2

2 2 4 83 4 12 364 4 16 64

5 9 45 2256 4 24 1447 11 77 5398 4 32 2569 6 54 48610 3 30 30011 1 11 12112 2 24 288

= 50 = 329 = 2467

i) Mean =

=

= 6.58

ii) Variance =

= -

= – (6.58)2

= 6.044

iii) Standard deviation =

=

= 2.458

b) If the number of tosses increases to 100

times, the mean will increase slightly.

Sum of the two

numbers ()

Frequency ( )

2

2 4 8 163 5 15 454 6 24 965 16 80 4006 12 72 4327 21 147 10298 10 80 6409 8 72 648

10 9 90 90011 5 55 60512 4 48 576

= 100 = 691

= 5387

Prediction of mean = 6.91

c) Test the prediction by determine the value of :

i. Mean

= 6.91

ii. Variance = -

= 2

= 6.122

iii. Standard deviation =

= 2.474

Prediction is proven.

a) Based on the table, the actual mean, the

variance and the standard deviation from

using the formula is:

Mean = x P(x)=

= 7Variance = x P(x) – (mean)=

- (7)2

= 54.83 – 49= 5.83Standard deviation = = 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.

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.

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)

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 n 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.

While I conducting the project, I had learned some moral values that I practice. This project had

taught me to responsible on the works that are given to me to be completed. This project also had make me felt more confidence to do works and not

to give up easily when we could not find the solution for the question. I also learned to be more

discipline on time, which I was given about two weeks to complete these project and pass up to my teacher just in time. I also enjoy doing this project during my school holiday as I spend my time with friends to complete this project and it

had tightens our friendship.

Objective…………………………..Part 1

Introduction……………………History……………………………..

Theoretical and empirical Probability............................

Part 2………………………………Part 3………………………………Part 4………………………………Part 5………………………………Further exploration………….

Reflection…………………………References…………………………Rubric………………………………