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PART 1
SHORT HISTORY OF PROBABILITY
“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.

PART 2

2 (a) Janet,Karen and Muna are playing the Monopoly game together. To start the
game, each player will have to toss the dice once. The player who obtains the highest
number will start the game.

Question : List all the possible outcomes when the die is tossed once.

Conjecture : All number on the dice will be appeared as the die is tossed once.

Steps : 1) Toss the die six times.
2) Record the number appeared each time.

Answer: {1,2,3,4,5,6}

2 (b) Instead of one die, two dice can 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″ appear on one
die and “3″ appears on the othr die, the outcome of the toss is (2,3). Hence, the
player shall move the token 5 spaces.

Question : 1) List all possible outcomes when two dice are tossed simultaneously.
2) Organize and present the list clearly by using table or chart.

Conjecture : For each number, there will be six possible outcomes.

Steps : 1) Take two dice of different base colour that is blue and white.
2) Toss the two dice simultaneously.
3) Record the dots on both turned-up faces for each toss.
4) Note that the events (2,3) and (3,2) should be treated as two different
events.

Answer : {(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)

Table 1
PART 3

Table 2 shows the sum of all dots on both turned-up faces when the two dice are tossed simultaneously

3 (a) Question : Complete table 1 by listing all possible outcomes and their corresponding probabilities

Answer :

Sum of the dots on both turned-up faces Possible Outcomes Probability, P(x)
2 {(1,1)} 1|36
3 {(2,1),(1,2)} 2|36
4 {(2,2),(3,1),(1,3)} 3|36
5 {(1,4),(4,1),(2,3),(3,2)} 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 2

3 (b) Question : Based on Table 2 that have been completed,
1) List all the possible outcomes of the following events
2) Find their corresponding probabilities

Answer :

A = { The two numbers are not the same }

{ (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) }

Probability : 5/6

B = { The products of the two numbers are greater than 36 }

{ }

Probability : 0

C = { Both numbers are prime or the difference between two numbers is odd }

{ (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) }

Probability : 1/2

D = { The sum of the two numbers are even and both numbers are prime }

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

Probability : 5/36