Probability - Probability Key Steps in Solving Problems on...

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Probability Key Steps in Solving Problems on Probability (1) Read the question very carefully, in particular, identify the following 2 major points: (a) how the experiment (action) is carried out? (b) what are the possible outcomes of the event? Note: For the event of getting 2 heads in tossing 3 coins, you should be careful that the outcome for ‘2 heads’ is ‘2 heads and 1 tail’. Moreover, you should remember that there are several combinations for ‘2 heads and 1 tail’: {HHT, HTH, THH}. This is just a simple example but most of the complicated problems will make use of the same tricks. (2) List the possible outcomes of the event. If the event consists of an infinite number of outcomes, you should still list some of them before deciding on the solving strategy. NOTE: UP TO THIS STEP. THE SOLVING STRATEGY IS NOT CONSIDERED. REMEMBER THAT THE SOLVING STRATEGY IS DETERMINED ONLY AFTER YOU HAVE IDENTIFIED ALL THE POSSIBLE OUTCOMES. (3) After you have fully understood the question, you can think about the solving strategy: (a) how to list the outcomes in your solution? There are several methods to be selected: -- Use set notation, e.g. tossing a die Sample Space={1,2,3,4,5,6} the time of finishing a test Sample Space={ t : min} 45 0 t -- Use an ordered pair to represent the outcome, e.g. tossing 2 dice. In this case, a coordinate system can be used to represent the outcomes as shown below: 6 5 4 3 2 1 1 2 3 4 5 6 AL APPLIED MATHEMATICS / PROBABILITY / P.1
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-- Use a Venn diagram, e.g. selecting a student who can be a member of Maths Club or Science Club or both. Math Science Club Club -- Use a tree diagram, e.g. select 2 balls from a bag consisting of red and blue balls red red blue red blue blue -- Other systematic ways to represent the outcomes (b) which method/formula to be used to find out the answer? IMPORTANT FORMULAE: n(S) n(A) P(A) = Addition Law: P(A B)=P(A)+P(B)-P(A B) P(A B)=P(A)+P(B) (if A and B are mutually exclusive) Multiplication Law: P(A B)=P(A)P(B|A) =P(B)P(A|B) P(A B)=P(A)P(B) (if A and B are independent) Conditional Probability: ) ( ) ( ) | ( B P B A P B A P = AL APPLIED MATHEMATICS / PROBABILITY / P.2
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Bayes’ Theorem: = = n j j j i i i A P A B P A P A B P B A P 1 ) ( ) ( ) ( ) ( ) | ( Sometimes, if the outcomes consists of different combinations, then the following formulae may have to be used: number of ways of arranging n unlike objects in a line n! number of ways of arranging n unlike objects in a line where p of one type are alike, q of another type are alike, r of a third type are alike, etc. !... ! ! ! r q p n number of ways of arranging n unlike objects in a ring when clockwise and anti-clockwise directions are different ( n -1)! number of permutations of r objects taken from n
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Probability - Probability Key Steps in Solving Problems on...

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