# SPM Form 5 Add Maths Project 2010 – Tugasan/Work 2

2### SPM Form 5 Additional Mathematic Project 2010 – Tugasan 2

### Additional Mathematic Project Work 2| 2010

- SPM Form 5 Add Math Project 2010

Copying is not encouraged. Just use the information here as a reference and then complete the work by yourself.

Part 1

The theory of probability has been applied in various fields such as market research, medical research, transportation, business, management and so on.

(a) Conduct research on the history of probability and give at least two examples on how the theory of probability is being applied in real life situations. Then, write an Introduction to this Project Work based on your findings. You may include the historical aspects, examples of the probability theory applications and its importance to real life situations.

(b) The probability theory can be divided into two categories: Theoretical Probabilities and Empirical Probabilities. Find out, discuss and write about the difference between the Theoretical and Empirical Probabilities.

Part 2

(a) Suppose you are playing the Monopoly game with two of your friends. To start the game, each player will have to toss the die once. The player who obtains the highest number will start the game. List all the possible outcomes when the die is tossed once.

(b) Instead of one die, two dice can also 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” appears on one die and “3” appears on the other, the outcome of the toss is (2, 3). Hence, the player shall move the token 5 spaces. Note: The events (2, 3) and (3, 2) should be treated as two different events.

List all the possible outcomes when two dice are tossed simultaneously. Organize and present your list clearly. Consider the use of table, chart or even tree diagram.

**Part 3**

Table 1 shows the sum of all dots on both turned-up faces when two dice are tossed simultaneously.

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

(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) I) = {The sum of the two numbers are even and both numbers are prime)

**Part 4**

(a) Conduct an activity by tossing two dice simultaneously 50 times. Observe the sum of all dots on both turned-up faces. Complete the frequency table below.

Based on Table 2 that you have completed, determine the value of:

(i) mean;

(ii) variance; and

(iii) standard deviation

of the data.

(b) Predict the value of the mean if the number of tosses is increased to 100 times.

(c) Test your prediction in (b) by continuing Activity 3(a) until the total number of tosses is 100 times. Then, determine the value of:

(i) mean;

(ii) variance; and

(iii) standard deviation

of the nibw data.

Was your prediction proven?

**Part 5**

When two dice are tossed simultaneously, the actual mean and variance of the sum of all dots on the turned-up faces can be determined by using the formulae below:

(a) Based on Table 1, determine the actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces by using the formulae given.

(b) Compare the mean, variance and standard deviation obtained in Part 4 and Part 5. What can you say about the values? Explain in your own words your interpretation and your understanding of the values that you have obtained and relate your answers to the Theoretical and Empirical Probabilities.

(c) If n is the number of times two dice are tossed simultaneously, what is the range of mean of the sum of all dots on the turned-up faces as n changes? Make your conjecture and support your conjecture.

**FURTHER EXPLORATION**

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. Conduct a research using the internet to find out the theory of LLN. When you have finished with your research, discuss and write about your findings. Relate the experiment that you have done in this project to the LLN.

**REFLECTION**

While you were conducting the project, what have you learnt? What moral values did you practise? Represent your opinions or feelings creatively through usage of symbols, illustrations, drawings or even in a song.

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SPM Form 5 Add Maths Project 2010 Work 2 [FULL VERSION] www.SPM2010.tk