UNIT 4 MATHEMATICAL METHODS SAMPLE REFERENCE MATERIALS

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1 UNIT 4 MATHEMATICAL METHODS SAMPLE REFERENCE MATERIALS EXTRACTS FROM THE ESSENTIALS EXAM REVISION LECTURES NOTES THAT ARE ISSUED TO STUDENTS Students attending our mathematics Essentials Year & Eam Revision Lectures in September and October will receive a comprehensive set Reference Materials for Eamination at NO ADDITIONAL CHARGE. In addition, students will receive a detailed list of the common errors made by students, prompts, eam tricks and watch-outs to ensure that valuable marks are not lost during the course of the eamination. Additional information and application forms for The Essentials lectures are available at

2 SOLVING WORDED PROBLEMS INVOLVING RATES OF CHANGE Step : Step : Step : Step 4: Step 5: Read the question carefully and identify the rate equation in question. Use the units associated with the variables to denote the type of rate in question. dv For eample: cm / min means that. dt If the rate equation has not be provided, use conventional formulae such as SOHCAHTOA, volume and area equations to establish an equation that contains your parameters. Count the number of variables present. If your equation is epressed in terms of more than one variable, use conventional formulae or any given data to epress the equation in terms of the required variable. Differentiate this equation to find the rate equation. Calculate the required parameter. Remember to state the appropriate units. WATCHOUTS Use units to assist in determining which rates are present or need to be found. dv For eample: cm / s represents. dt If a quantity is increasing in value, then the corresponding rate is positive. If a quantity is decreasing in value, then the corresponding rate is negative. Always determine whether the given equation represents a rate equation or otherwise. If the equation represents a rate, then rates are evaluated by substituting values into the given equation. Otherwise, equations will need to be differentiated or integrated to obtain a rate epression. If question asks for the rate at which something is decreasing, the answer must be positive (as the word decreasing takes into account that the answer is negative). dv For eample: If ml / s then the rate at which the volume is decreasing is dt ml / s NOT ml / s. The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page

3 RELATIONSHIPS BETWEEN DISPLACEMENT, VELOCITY AND ACCELERATION (when they are epressed as functions of time) Displacement () Differentiate d dt Anti-differentiate ( v dt) Velocity (v) Differentiate dv dt Anti-differentiate ( a dt) Acceleration (a) For eample: Displacement: Velocity: Acceleration: t 5 h 5 e 65t dh t t e 65 e 65 m/ s dt 5 dv t t 5 5 e 5 e m/ s dt 5 The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page

4 CHOICES AND COMBINATIONS OF EVENTS In many probability applications, the required answer may consist of either: A choice of events: Such questions incorporate the word OR, or imply that any of the listed events may or are to occur. A combination of events (multiple events): Such questions incorporate the word AND, or imply that the combination of the listed events may or are to occur. Events cannot occur at the same time (simultaneously) CHOICES OF EVENTS OR (+) (Use Addition Rule) Events can occur at the same time (simultaneously) Events Mutually Eclusive Pr( A B) Pr( A) + Pr( B) Events Not Mutually Eclusive Pr( A B) Pr( A) + Pr( B) Pr( A B) Outcome of one event is dependent on the outcome of the previous event (conditional) MULTIPLE EVENTS AND (X) (Use Multiplication Rule) Outcome of one event is not dependent on the outcome of the previous event Events Dependent ( A B) ( A B) ( B) Pr Pr.Pr Events Independent Pr( A B) Pr( A) Pr( B) The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 7

5 For eample: The probability of drawing a card which is a or a King. These are mutually eclusive events as a card cannot be a and a King at the same time. Therefore: 4 4 Pr( or King) Pr() + Pr( King) For eample: The probability of drawing a card which is a heart or a King. These are not mutually eclusive events as a card can be both a heart and a King. Pr( Heart or King) Pr( Heart) + Pr( King) Pr( Heart and King) For eample: The probability of drawing a and a King (with replacement). These are independent events where order has not been specified. Every possible order in which events can occur must be taken into consideration. Therefore: [ ] [ ] Pr( and King) Pr() Pr( King) + Pr( King) Pr() For eample: The probability of drawing a and then a King (with replacement). These are independent events where order has been specified. Therefore, we only consider the given order: Pr( and King) Pr() Pr( King) The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 7

6 Eample: Two events, P and R are such that Pr( P R) Pr( P) + Pr( R) Pr( P R) Pr( P ), Pr( R ) and 5 Pr( P R). Find Pr( P R). 4 + Pr( P R ) Pr( P R) 6 Eample: Given that Pr( A ). 45, Pr( B ). 58 and Pr( A B). 8, find Pr( A / B). Pr.8 ( A B) Pr( A) + Pr( B) Pr( A B) Pr ( A B) Pr. ( A / B) ( A B) ( A B). Pr( B) Pr Pr WATCHOUTS When working with multiple events, you must take the order of the events into consideration. For eample: A coin is tossed twice. The probability that one of the faces that appears is a tail is given by: Pr( one tail) Pr( Tail AND Head) OR Pr( Head AND Tail) + For eample: If the probability that Andrew remembers to take out the garbage on garbage night is.7, while the probability that Mark remembers to take out his garbage is.4 the probability that one, not both, will remember to take out the garbage out is given by: \ Pr( One takes garbage out) Pr( Andrew does AND Mark does not) OR Pr( Mark does AND Andrew does not ) Pr( One takes garbage out ) (.7.6) + (.4.).54 The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 7

7 CONDITIONAL PROBABILITY Often, when dealing with probabilities, we are concerned with some but not all of the outcomes of an eperiment. Introducing such conditions has the effect of reducing the set of possible events, and hence the probability of an outcome. To calculate the probability of event A occurring given that event B has already occurred, we apply the following rule: Pr ( A B) Pr ( A B) Pr( B) Common elements to both events What you want Condition WATCHOUTS Use the conditional probability rule when you are concerned with some but not all of the outcomes of an eperiment or when a condition has been introduced that reduces the total number of possible outcomes. Key words in conditional probability If / Given that / Providing that / Hence Eam writers often omit the key indicator words for conditional probability. Therefore, with each part of a question, ask yourself whether the total number of outcomes has decreased. When applying the rule DO NOT reduce the number of possible outcomes. Use the probabilities in the unrestricted scenario. For eample: Pr( heart AND red) Pr( heart red) Pr( red) At times, questions can be quickly solved using logic or reasoning. For eample: hearts Pr( heart / red) 6 red The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 75

8 WATCHOUTS If eperiencing difficulties working out what the intersection of the domains is, use a number line (draw the appropriate intervals and write down the areas common to both functions). Pr ( X X < ) Pr( X X < ) Pr( X < ) Pr( X < ) Pr( X < ) ( X X ) Pr > Pr( X X > ) Pr( X > ) Pr( X Pr( X ) > ) We can also solve conditional probability applications using common sense. If using common sense, you must reduce the sample size to take into consideration the introduced restriction. For eample: Find the probability of throwing a 4 on a die given that an even number has appeared. Since we are only interested in the outcomes, 4 and 6 (even number) then there are only possible outcome. Therefore, the sample size needs to be reduced from 6 to. Pr(4 even number) If the formula is to be used then use the whole sample size which is 6: By Rule: Pr(4 and even) Pr( 4 even ) 6 Pr( even) 6 If two events A and B are independent (i.e. the probability of A occurring does not depend on B occurring) then Pr( A B) Pr( A) Pr( B). Therefore: Pr( A / B) Pr(A ) in this case. The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 76

9 TYPES OF PROBABILITY DISTRIBUTIONS BEING ADDRESSED AT VCE LEVEL This study will concentrate on the following probability distributions: General Discrete Distributions Binomial Distributions (Bernoulli Sequences) Markov Sequences General Continuous Distributions Normal Distributions WATCHOUTS A question that asks to find the probability of a specific outcome cannot be a continuous probability distribution. Eg. Pr( X 5). Analysis questions typically change distributions throughout the task. Therefore, before you attempt each part of a question, ask yourself whether or not the distribution has changed. With each new part of a question, ask yourself whether the type of distribution is the same, but the definitions of the parameters have changed eg. X no. of failures in part (a) and then part (b) asks for number of successes. If a question indicates that the variable is discrete, do not automatically assume that you are required to apply general discrete applications. You may in fact be required to apply binomial rules or Markov sequences! To prove that something represents a probability distribution function, show that the sum of the probabilities of all the possible outcomes is equal to and that each probability lies between and. The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 9

10 THE BINOMIAL (BERNOULLI) DISTRIBUTION There are many occasions where the same trial is repeated a number of times, and there are just TWO possible outcomes in each trial. For eample a pass or fail, male or female, a success or failure. A Bernoulli Sequence is the name used to describe a sequence of identical trials that display the following characteristics: The events have two possible outcomes A success ( p ) and a failure ( p) (or q ). The eperiment consists of a set number of repeated trials ( n ). These trials are identical. Trials are independent i.e. the outcome of one trial has no influence on the outcome of any other similar trial. The probability of a success is the same in each trial (eg. Sampling with replacement). The number of successes in a Bernoulli sequence of n trials is called a binomial random variable and is said to have a binomial probability distribution. If X represents a random variable which has a binomial distribution then it may be epressed as: X Bi n, p X ~ B n, p ~ ( ) or ( ) where: n represents the number of trials. p represents the probability of success. Note: Bernoulli sequences are usually resolved using tree diagrams or binomial probability distribution applications. This distribution is used to solve applications involving discrete random variables. The eperiment consists of more than one trial. The trial size will usually be stated. The trial size (or sample size) is constant, and will not change. There are only two possible outcomes for each trial: A success and a failure. Use when sampling with replacement. The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 4

11 CALCULATING PROBABILITIES The probability of number of successes in n trials of a binomial eperiment is determined using the following rule: Pr n ( X ) ( ) p ( p) ( ) n! n n! ( n ) no. of successes no. of failures n For eample: Pr(X5) (probability of success) (probability of failure) 5 Note: X represents the binomial random variable eg. the number of successes. n represents the trial size. p represents the probability of a success. This formula can only be applied if order is NOT IMPORTANT. If order is important, the multiplication principle will need to be applied.! n C TI-NSPIRE CAS binompdf (n, p, ) binomcdf (n, p, -min, -ma) For eample: Find Pr( X ) if X Bi(,.5). Pr( X ) binompdf (,.5,) For eample: Find Pr( X < ) if X Bi(,.5). Pr( X < ) binomcdf (,.5,,) CASIO CLASSPAD BinomialPDf (, n, p) BinomialCDf (-ma, n, p) For eample: Find Pr( X ) if X Bi(,.5). Pr( X ) binompdf (,,.5) For eample: Find Pr( X < ) if X Bi(,.5). Pr( X < ) binomcdf (,,.5) binomcdf (9,,.5) For eample: Find Pr( X ) if X Bi(,.5). Pr( X ) binomcdf (,,.5) binomcdf (9,,.5) The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 5

12 WATCHOUTS Something that is typically a binomial application can be solved using discrete applications and visa versa. Which technique is used depends upon the manner in which the question has been phrased. X and p must represent the same outcome. Eg. If X represents the number of cures then p must represent the probability of a cure. Always let X equal the topic of the question (usually stated at the end of the sentence). This formula can only be applied if order is NOT IMPORTANT. For eample: Find the probability that 5 questions are correctly answered. If order is important, the multiplication principle will need to be applied. For eample: Find the probability that the first 5 questions are correctly answered. Watch out for changes in the definition of X or p. You may need to re-define X and p throughout a question. For eample: Given the Pr(success) and you are asked for the probability of a certain number of failures. Changes in the definition of success ( p changes). Take care when using previously calculated values eg. Part (b) asks that you calculate probability that you miss, Part (c) requires probabilities associated with hits. The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 6

13 CALCULATING THE MEAN AND STANDARD DEVIATION Epected Value: E ( X ) np Variance of X: Var ( X ) np( p) As q Var ( X ) npq p Standard Deviation: σ npq μq μ( p) For Eample: A coin is tossed 6 times. (a) Find the mean of the number of heads that will occur. n 6, p.5, q.5.5 μ np 6(.5) (b) Find the standard deviation of the number of heads that will occur. State your answer correct to decimal places. σ np( p) 6(.5)(.5) 5.5 (c) Hence find a 95% confidence interval for the number of heads that will occur. State your answers as whole numbers. 95% confidence interval: Pr( μ σ X μ + σ ). 95 (.5) X + (.5) 75.5 X X 4 The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 7

14 CALCULATING PROBABILITIES OF SIMPLE PDFS Probabilities associated with continuous probability functions are determined by integration. Pr( a X b) f ( ) d As the entire probability function must be located above the X ais (probabilities i.e. areas cannot be negative), when finding areas or definite integrals there is no need to be concerned about writing separate integrals for each of the regions that lie above and below the X ais. Simply substitute in the relevant upper and lower limits into the antiderivative(s). The only thing students need to worry about is whether there are different equations for different parts of the domains. b a For eample: f ( ) ( ) Otherwise y X ( ) Pr(.5) d. 5 Note that the lower limit is (as per given domain). X ( ) Pr( >.75).75 d. 67 Note that the upper limit is (as per given domain)..75 Pr(.5 X <.75) ( ) d Pr( X >.) Pr( X >.5) Pr( X >. X >.5) Pr( X >.5) Pr( X >.) Pr( X >.5).9995 The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page

15 PROBABILITIES INVOLVING COMPLEX PDFS If asked to find a probability of a continuous probability distribution that is defined by or more equations other than f ( ), simply calculate the area under each corresponding curve and add your results. For Eample: For the probability density function given by: 4 f ( ) 4 < < 4 < or > 4 Pr( X < ) d + 4 d f() f() Pr( X > ) d + d or 4 4 Pr( X ) d 4 f() f() If the probability distribution is defined by an absolute value function, you will need to adjust domains to take into account the upper and lower limits of each curve. The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page

16 THE MEAN OF SIMPLE PDFS The mean or epected value of a continuous random variable X is given by: i.e. μ f ( ) d, provided the integral eists. i.e. Take equation f (), multiply it by, then integrate with respect to before substituting in the upper and lower limits. For eample: f ( ) ( ) Otherwise y μ f( ) d 4 ( ) ( ) d 5 4 d WATCHOUTS Although the rule describing the mean requires that we integrate between ±, evaluate integrals over the interval for which the function is defined. At times, you will be able to use the symmetry properties of a curve to determine the mean. If the curve shows perfect symmetry about some vertical line, then the value of the vertical line represents the mean. Do not use symmetry properties if asked to use calculus to find the mean. You will need to use integration in these cases. To show that the mean is equal to some value substitute the value of the mean into μ f ( ) d and solve. Given an absolute value function you will have to adjust the domains from f ( ) and f ( ) < to take into account the domains across which the probability density function is defined. The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page

17 THE MEAN OF COMPLEX PDFS If asked to find the mean of a continuous probability density function that is defined by or more equations other than f ( ), simply apply the rule f ( ) d on each individual section of the graph and add your results. For eample: y 4 f ( ) 4 < < 4 < or > μ d d d + d If the probability distribution is defined by an absolute value function, you will need to adjust domains to take into account the upper and lower limits of each curve. For eample: k( ) 6 f ( ) becomes > 6 or < k( ) f( ) k( ) Elsewhere The School For Ecellence 5 The Essentials Mathematical Methods Reference Materials Page 4