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ADDITIONAL MATHEMATICS PROJECT WORK 2 2010

PROJECT WORK FORADDITIONAL MATHEMATICS

2010

Probability in Our Life

NAME : CHONG LI LIAN

TEACHER : PN.

I/C : 931027-06-5038SCHOOL : SMK TENGKU AFZAN

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CONTENT

No Contents Page

1 Introduction

2 Part 1

3 Part 2

4 Part 3

5 Part 4

6 Part 5

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INTRODUCTION

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 areas of study as mathematics, statistics, finance, gambling, science, and

philosophy to draw conclusions about the likelihood of potential events and the underlying

mechanics ofcomplex systems.

Interpretations

The wordprobabilitydoes not have a consistent directdefinition. In fact, there are sixteenbroad categories ofprobability interpretations, whose adherents possess different (and

sometimes conflicting) views about the fundamental nature of probability:

1. Frequentists talk about probabilities only when dealing with experiments that arerandom and well-defined. The probability of a random event denotes the relative

frequency of occurrence of an experiment's outcome, when repeating the

experiment. Frequentists consider probability to be the relative frequency "in the

long run" of outcomes.[1]

2. Bayesians, however, assign probabilities to any statementwhatsoever, even whenno random process is involved. Probability, for a Bayesian, is a way to represent anindividual'sdegree of beliefin a statement, or an objective degree of rational belief,

given the evidence.

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PART 1

a) History of probability

The scientific study of probability is a modern development. Gambling shows that there has beenan interest in quantifying the ideas of probability for millennia, but exact mathematicaldescriptions of use in those problems only arose much later.

According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable'

(Latinprobabilis) meant approvable, and was applied in that sense, univocally, to opinion and toaction. A probable action or opinion was one such as sensible people would undertake or hold, in

the circumstances. However, in legal contexts especially, 'probable' could also apply topropositions for which there was good evidence.

Aside from some elementary considerations made by Girolamo Cardano in the 16th century, the

doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal(1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.

Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine ofChances (1718) treated the subject as a branch of mathematics. See Ian Hacking's The

Emergence of Probability and James Franklin's The Science of Conjecture for histories of theearly development of the very concept of mathematical probability.

The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous,1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory

to the discussion of errors of observation. The reprint (1757) of this memoir lays down theaxioms that positive and negative errors are equally probable, and that there are certain

assignable limits within which all errors may be supposed to fall; continuous errors are discussedand a probability curve is given.

Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of

observations from the principles of the theory of probabilities. He represented the law ofprobability of errors by a curvey = (x), x being any error and y its probability, and laid down

three properties of this curve:

1. it is symmetric as to they-axis;2. thex-axis is an asymptote, the probability of the error being 0;3. the area enclosed is 1, it being certain that an error exists.

He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but

one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle ofthe maximum product of the probabilities of a system of concurrent errors.

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PART 1

The method of least squares is due to Adrien-Marie Legendre (1805), who introduced it in hisNouvelles mthodes pour la dtermination des orbites des comtes (New Methods for

Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American

writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,

h being a constant depending on precision of observation, and c a scale factor ensuring that the

area under the curve equals 1. He gave two proofs, the second being essentially the same as John

Herschel's (1850). Gauss gave the first proof which seems to have been known in Europe (thethird after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823),

James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W. F. Donkin (1844, 1856),and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher

(1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula forr, the probable error of asingle observation, is well known.

In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix

(1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872),Hermann Laurent (1873), Liagre, Didion, and Karl Pearson. Augustus De Morgan and GeorgeBoole improved the exposition of the theory.

Andrey Markov introduced the notion ofMarkov chains (1906) playing an important role in

theory ofstochastic processes and its applications.

The modern theory of probability based on the meausure theory was developed by AndreyKolmogorov (1931).

On the geometric side (see integral geometry) contributors to The Educational Times wereinfluential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).

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PART 1

b) Probability in our livesi) 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 Servicewhen they look at all other days in their historical database that have the same weathercharacteristics (temperature, pressure, humidity, etc.) and determine that on 70% of similar days

in the past, it rained.

As we've seen, to findbasic 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 ofsimilar 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 thesedays 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 greaterthan 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 anevent 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 averagesLet'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 favorableoutcome. 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 300batting average.

This means that when a Major Leaguer with a batting average of 300 steps up to the plate, he hasonly a 30% chance of getting a hit - and since most batters hit below 300, you can see how hardit is to get a hit in the Major Leagues!

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PART 1

c) Difference between the Theoretical and Empirical ProbabilitiesThe 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 howoften the event occurs after collecting data or running an experiment (in a large number oftrials). 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.

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 observedprobability) 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 expectedtheoretically.

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PART 3

a) Table 1 show the sum of all dots on both turned-up faces when two dice are tossed

simultaneously.

Sum of the dots onboth 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

Table 1

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PART 4

a)

Sum of the two

numbers ()

Frequency () 2

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

= 50 = 329 Table 2

i) Mean =

=

= 6.58

ii) Variance == -

= (6.58)

2

= 6.044

iii)Standard deviation =

= = 2.458

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PART 4

b)

Sum of the two

numbers ()

Frequency () 2

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

= 100 = 691

Prediction of mean = 6.91

i. Mean

= 6.91

ii. Variance = - = 2

= 6.122

iii. Standard deviation = = 2.474

Prediction is proven.

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PART 5

a)

Mean = x P(x)

=

= 7

Variance = x2P(x) (mean)

2

=

- (7)2

= 54.83 49

= 5.83

Standard deviation =

= 2.415

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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

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 andstandard deviation.

b. Law of Large Number is not an absolute law. Violation of this law is still possible thoughthe 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.

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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.

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)

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