Additional Mathematics Project Work

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Table of ContentsPart 1...................................................................................................................................................2

a)..............................................................................................................................................................2

b).........................................................................................................................................................2

Part 2......................................................................................................................................................3 - 4

a)..............................................................................................................................................................3

b).........................................................................................................................................................4

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Part 1a) Probability theory is the branch of mathematics concerned with analysis of random

phenomena.

Probability theory was originally inspired by gambling problems. The earliest work on the subject was performed by Italian mathematician and physicist Girolamo Cardano (1501–1576). In his manual Liber de Ludo Aleae, Cardano discusses many of the basic concepts of probability complete with a systematic analysis of gambling problems. Unfortunately, Cardano's work had little effect on the development of probability because his manual did not appear in print until 1663—and even then received little attention.

In 1654, another gambler named Chevalier de Méré invented a system for gambling that he was convinced would make money. He decided to bet even money that he could roll at least one twelve in 24 rolls of two dice. However, when the Chevalier began losing money, he asked his mathematician friend Blaise Pascal (1623–1662) to analyze his gambling system. Pascal discovered that the Chevalier's system would lose about 51 percent of the time.

Pascal became so interested in probability that he began studying more problems in this field. He discussed them with another famous mathematician, Pierre de Fermat (1601–1665) and, together they laid the foundation of probability theory.

Insurance policies are made possible by empirical probability. We know the amount of accidents, and we know the amount of times something happens without error. Based on that, it can be calculated what the chance (and thus the cost) is of a certain event.

Gambling (professional) 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.

Dangerous medical procedures can 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.

b) Empirical probability is anything where the probability has been determined by repeated attempts.Theoretical probability is anything where the probability has been calculated using predetermined variables used as a model.

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For example, if we were doing a lab related to acceleration due to gravity, the theoretical value would be 9.81 m/s2 and the empirical value would be the value we calculated.

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Part 2a) List of all the possible outcomes when the die is tossed once

= {1,2,3,4,5,6}

b) List of all the possible outcomes when two dice are tossed simultaneously:

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2

2

2

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Part 3a) Table 1 shows the sum of all dots

on both turned- up faces when two dice are tossed simultaneously

Sum of the dots on both 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

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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) 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}

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

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)

30/36

B 0/30

C (5,5),(5,3),(5,2),(3,5),(3,3),(3,2),(2,5),(2,3),(2,2),(6,5),(6,3),

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

20/36

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D (5,3),(5,5),(3,3),(3,5),(2,2) 5/36

Table 2

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Part 4

Sum of the two numbers (x) Frequency (f)

2

3

4

5

6

7

8

9

10

11

12

2

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Referenceshttp://en.wikipedia.org/wiki/Probability_theory

http://www.scienceclarified.com/Ph-Py/Probability-Theory.html

http://science.jrank.org/pages/5505/Probability-Theory-Empirical-probability.html