SPM Form 5 Add Maths Project 2010 – Tugasan/Work 2 [FULL ANSWER]a
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SPM Form 5 Add Maths Project 2010 – Tugasan/Work 2 [ANSWER]
PART 1
a) History of probability
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.
According to Richard Jeffrey, “Before the middle of the seventeenth century, the term ‘probable’ (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. 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 to propositions 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 of Chances (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 the early 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 the axioms 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 discussed and a probability curve is given.
PierreSimon 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 of probability of errors by a curve y = φ(x), x being any error and y its probability, and laid down three properties of this curve:
 it is symmetric as to the yaxis;
 the xaxis is an asymptote, the probability of the error being 0;
 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 of the maximum product of the probabilities of a system of concurrent errors.
PART 1
The method of least squares is due to AdrienMarie Legendre (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre’s contribution, an IrishAmerican 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 (the third 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 for r, the probable error of a single 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 George Boole improved the exposition of the theory.
Andrey Markov introduced the notion of Markov chains (1906) playing an important role in theory of stochastic processes and its applications.
The modern theory of probability based on the meausure theory was developed by Andrey Kolmogorov (1931).
On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).
PART 1
a) 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 atbats a player has had, not including walks. A hit is a favorable outcome. Thus if in 10 atbats 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
a) Introduction
PART 1
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? 


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. 
PART 2
a) {1, 2, 3, 4, 5, 6}
b)
PART 3
a) Table 1 show the sum of all dots on both turnedup faces when two dice are tossed simultaneously.
Sum of the dots on both turnedup 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
PART 3
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) }  
B  ø  ø 
C  P = Both number are prime
P = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)} Q = Difference of 2 number 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) } 

D  P = Both number are prime
P = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)} 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 ∩ R D = {(2,2), (3,3), (3,5), (5,3), (5,5)} 
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  = 2467  
Table 2
i) Mean =
=
= 6.58
ii) Variance =
= –
= – (6.58)^{2}
= 6.044
iii) Standard deviation =
=
= 2.458
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  = 5387  
Prediction of mean = 6.91
 Mean
= 6.91
 Variance = 
=^{2}
= 6.122
 Standard deviation =
= 2.474
Prediction is proven.
PART 5
a)
Mean = x P(x)
=
= 7
Variance = xP(x) – (mean)
^{= }^{ } (7)^{2}
= 54.83 – 49
= 5.83
Standard deviation =
= 2.415
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
 The sample (n=100) is not large enough to see the change of value of mean, variance and standard deviation.
 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.
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)
SPM Form 5 Add Maths Project 2010 Work 2 [FULL VERSION] www.SPM2010.tk
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