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Dear 4E/5N Students You are expected to do the following for the Maths module:

General Instructions. Dear 4E/5N Students You are expected to do the following for the Maths module: Read and study the slides on Probability Explore the websites given Complete the one task given on the worksheet and submit them to your math teacher on 5 February 2008.

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Dear 4E/5N Students You are expected to do the following for the Maths module:

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  1. General Instructions Dear 4E/5N Students You are expected to do the following for the Maths module: • Read and study the slides on Probability • Explore the websites given • Complete the one taskgiven on the worksheet and submit them to your math teacher on 5 February 2008. Happy Enjoy-Learning! Best Regards Your Math Teachers

  2. Probability

  3. Introduction • In this e-lesson, you will learn to • solve simple probability problems inPart One • use possibility diagrams and tree diagrams to solve probability problems involving combined events inPart Two

  4. Introduction Probability Theory was first used to solve gambling problems. Lotteries have always been a magnet to those who dream of instant riches. Toto, 4-D and Singapore Sweep are some of the favourite games of chance among Singaporeans. However, do you know that you have aone-in-8.1 millionchance of winning the first prize for Toto, and the odds for striking any of the first three prizes in the Singapore Sweep and 4-D areone in 3 million and one in 10,000respectively? Do you know how to calculate the odds?

  5. Introduction Probability Theory has since been widely used in areas like business, finance, science and industry, and has become a powerful branch of mathematics. We often make statements involving probability orchancein our daily life. Some examples of these statements are: ‘It will probably rain today.’ ‘It is unlikely that we will win the championship.’ ‘There is a high chance that you will find him in the canteen.’ ‘It is impossible to pass the test!’

  6. Part One

  7. Experiments Probabilityis defined as thelikelihoodof an occurrence of a special event. In probability, anexperiment is an operation or a process with a result or an outcome whose occurrence depends onchance. Some examples of an experiment are: Example 1Tossing a coin Example 2Tossing a dice

  8. Sample Space An experiment can result in several possible outcomes. The set of all possible outcomes is called thesample spaceor probability space, S. Example 1 Tossing a coin Possible Outcomes: Head or Tail  S = {Head, Tail} Example 2 Tossing a dice Possible Outcomes: 1 or 2 or 3 or 4 or 5 or 6  S = {1, 2, 3, 4, 5, 6}

  9. Events An event, Eis a particular result of an experiment. Hence,Econtains some or all of the possible outcomes inS. Example 1 Tossing a coin Let E be the event of getting a tail.  E = {Tail} Example 2 Tossing a dice Let E be the event of getting a number less than 5 on the dice.  E = {1, 2, 3, 4}

  10. Simple Probability The probability of an event E occurring is given by where n(E) is the number of outcomes in E and n(S) is the total number of possible outcomes in S.

  11. Simple Probability Example 1 Tossing a coin S = {Head, Tail} and n(S) = 2 E = {Tail} and n(E) = 1 Example 2Tossing a dice S = {1, 2, 3, 4, 5, 6} and n(S) = 6 E = {1, 2, 3, 4} and n(E) = 4

  12. Simple Probability • The probability of any event occurring lies between • 0 and 1 inclusive, i.e.0 ≤ P(E) ≤ 1. • Do you know why? • Some important notes: • If P(E) = 0, then the event cannot possibly occur. • If P(E) = 1, then the event will certainly occur. • Probability of an event E not occurring • = 1 – probability of an event occurring • i.e.P(E’) = 1 – P(E)

  13. Sample Question Question: In an experiment, a card is drawn from a pack of 52 playing cards. (a) What is the total number of possible outcomes of this experiment? (b) What is the probability of drawing (i) a black card, (ii) a green card, (iii) a red ace, (iv) a heart, (v) a card which is not a heart?

  14. Solution to Sample Question (a) Total number of possible outcomes, n(S) = 52. (b)(i) P(drawing a black card) = (ii) P(drawing a green card) = (iii) P(drawing a red ace) =

  15. Solution to Sample Question (b)(iv) P(drawing a heart) = (v) P(drawing a card which is not a heart) = 1 – P(drawing a heart)

  16. Part Two

  17. Possibility Diagrams and Tree Diagrams Possibility diagrams and tree diagrams are used to list all possible outcomes of a sample space in a systematic and effective manner. These diagrams are useful for finding the probabilities of combined events.

  18. H T 2nd coin Eg. P(getting 2 heads) = P(HH) = H T An example of possibility diagrams Two coins are tossed together. The possibility diagram below shows all the possible outcomes: S = { } Each represents an outcome. HH, HT, TH, TT 1st coin

  19. An example of tree diagrams Probability Outcome HH HT TH TT 1st coin 2nd coin P(HH) = ½  ½ = ¼ ½ H H ½ T ½ P(HT) = ½  ½ = ¼ P(TH) = ½  ½ = ¼ ½ H ½ T T ½ P(TT) = ½  ½ = ¼ Two coins are tossed together. The tree diagram below shows all the possible outcomes: Each outcome is obtained by tracing along a branch from left to right. The probability of each outcome is obtained bymultiplyingthe probabilities along the respective branch. The total probability of all possible outcomes is¼+¼+¼+¼ = 1.

  20. Sample Question Question: A box contains three cards numbered 1, 3, 5. A second box contains four cards numbered 2, 3, 4, 5. A card is chosen at random from each box. (a) Show all the possible outcomes of the experiment using a possibility diagram or a tree diagram. (b) Calculate the probability that (i) the numbers on the cards are the same, (ii) the numbers on the cards are odd, (iii) the sum of the two numbers on the cards is more than 7.

  21. 5 2nd box 4 3 2 1 3 5 1st box Solution to Sample Question (a) Using a possibility diagram: S = { (1,2), (1,3), (1,4), (1,5), (3,2), (3,3), (3,4), (3,5), (5,2), (5,3), (5,4), (5,5) } n(S) = 12 (b)(i) P(both numbers are the same) = (b)(ii) P(both numbers are odd) = (b)(iii) P(sum > 7) =

  22. 2 2 2 3 3 3 4 4 4 5 5 5 1 3 5 Solution to Sample Question (a) Using a tree diagram: (b)(i) P(both numbers are the same) = P[(3,3) or (5,5)] = 1st box 2nd box (b)(ii) P(both numbers are odd) = P[(1,3) or (1,5) or (3,3) or (3,5) or (5,3) or (5,5)] = (b)(iii) P(sum > 7) = P[(3,5) or (5,3) or (5,4) or (5,5)] =

  23. Websites: http://mathforum.org/dr.math/faq/faq.prob.intro.htmlhttp://regentsprep.org/Regents/math/math-a.cfm#a6http://www.bbc.co.uk/schools/ks3bitesize/maths/handling_data/index.shtml

  24. Assignment Task: Complete the Multiple Choice Questions. Do your work on foolscap paper and show your working clearly. NOTE: Submit your assignment to your Math Teacher on 05 February 2008.

  25. References:1. Lee, P. Y., Fan, L. H., Teh, K. S. and Looi, C. K. (2002) New Syllabus Mathematics 4 Singapore: Shing Lee Publishers Pte Ltd.2. Tay, C. H. (2003) New Mathematics Counts for Secondary 5 Normal (Academic) Singapore: Federal Publications.

  26. End of e-Lesson

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