MMC Math 2009

transcript

Relationship and Applications of MATHEMATICS

OutlineOutline Selected topics review

1. PROBABILITY2. POWERS AND EXPONENTS3. LINEAR EQUATIONS4. END

PROBABILITYPROBABILITYLEARNING OUTCOMESTOPICEXAMPLES

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LEARNING OUTCOMESLEARNING OUTCOMES At the end of the class, students should be

able to:1. Determine whether an outcomes is a possible

outcome of an experiment.2. Determine the sample space of an experiment.3. Write the sample by using set notation.4. Identify the elements of a sample space which satisfy

given condition.5. Find the ratio of the number of times an event occurs

to the number of trials.

PROBABILITY PROBABILITY MENUMENU

1) PROBABILITY 1) PROBABILITY

EVENTEVENT

SAMPLE SPACESAMPLE SPACE

PROBABILITY OF EVENTPROBABILITY OF EVENT

PROBABILITY PROBABILITY MENUMENU

PROBABILITY IPROBABILITY I1.1 SAMPLE SPACE1.1 SAMPLE SPACE

TOPIC MENUTOPIC MENU

SAMPLE SPACESAMPLE SPACEAn experimentexperiment is a process or an

operation with an outcomes.

Toss a balanced die once and observe its uppermost face.

SAMPLE SPACESAMPLE SPACE When toss the coin, we can get only 2

results:

1.1. HeadHead2.2. TailTail

SAMPLE SPACESAMPLE SPACE The set of all possible outcomes of an experiment is

called the sample spacesample space. It usually denoted by SS.

Example 1:

En. Adam has a fruit stall that sells bananas, apples, watermelons, papayas and durians. Students of class 4KP are asked to select their favorite fruit from the fruits at En. Adam’s stall.

S S = { banana, apple, watermelon, papaya, durian}

SAMPLE SPACESAMPLE SPACE Example 2:

A month is randomly selected from a year. Describe the sample space of this experiment by using set notation.

SS= { January, February, March, April, May, June, July, August, September, October, November, December}

PROBABILITY PROBABILITY 1.2 EVENT1.2 EVENT

EVENTEVENT Is a subset of the sample space. Is an outcome or a set of outcomes

that satisfies certain condition. Denoted by a capital letter.

EVENTEVENT Example 1:

A box contains five cards written with 1,2,3,4 and 5 respectively. A card is picked randomly from the box.

S = {1, 2, 3, 4, 5}. If we define J as ‘the card with J as ‘the card with

an even number’an even number’,, the outcome of J in set notation will be

J = { 2, 4 }.J = { 2, 4 }.

J is known as an event of the experiment.

The number of outcome of an event n(P)=2

EVENTEVENT Example 2:

A letter is randomly selected from the word ‘COMPUTER’. Determine the number of possible outcomes of the event that the selected letter is

i. A vowel ii. A consonant

SolutionSolutioni. Let A = event that the selected letter is vowel = {O, U, E}

Therefore n (A) = 3

ii. Let B = event that the selected letter is consonant = {C, M, P, T, R} Therefore n (B) = 5

PROBABILITY PROBABILITY 1.3 PROBABILITY OF AN EVENT1.3 PROBABILITY OF AN EVENT

PROBABILITY OF AN EVENTPROBABILITY OF AN EVENT

Probability of an event E,

P(E) = number of outcomes of the eventP(E) = number of outcomes of the eventnumber of outcomes of thenumber of outcomes of the sample spacesample space

P(E) = n (E)P(E) = n (E) n (S)n (S)

0 ≤ P(E) ≤ 10 ≤ P(E) ≤ 1

PROBABILITY OF AN EVENTPROBABILITY OF AN EVENT

P(E) = 0P(E) = 0 means that it is impossible for the event to happen.

P(E) =1P(E) =1 means that the event is certain to happen.

The closer the probability of a given event is to 1, the more likely it is to happen.

PROBABILITY OF AN EVENTPROBABILITY OF AN EVENT ExampleExample

A bag contains 3 red balls and 4 white ones. If Rashid puts his hand in the bag and picks a ball, what is the probability that the ball he picked is white?

Solution:Solution:

S = {R1,R2,R3,W1,W2,W3,W4}n(S)= 7

Let E is the event of drawing a white ballE = {W1,W2,W3,W4}n(E)=4

Therefore, the probability of drawing a white ball is 4 7

PROBABILITY PROBABILITY EXAMPLESEXAMPLES

EXERCISE 1EXERCISE 1

A number from 1 to 11 is chosen at random. What is the probability of choosing an odd number?

A. 1/11B. 5/11C. 6/11D. None of above

TUTORIAL TUTORIAL

EXERCISE 2EXERCISE 2 A bag consists of 3 green, 1 white and 1

purple chips. Two chips are drawn from the bag. Which of the following outcomes are possible?

A. (green, red)B. (green, green)C. (purple, purple)D. (white, white)

TUTORIALTUTORIAL

EXERCISE 3EXERCISE 3 A dice is rolled 420 times. How many

times will a number greater than 4 occur?

A. 70B. 140C. 210D. 360

TUTORIALTUTORIAL

EXERCISE 4EXERCISE 4 There are 45 boys and girls in a class.

Given the probability that a boy is chosen is 4/15. the number of girls is

A. 8B. 12C. 25D. 33

TUTORIALTUTORIAL

EXERCISE 5EXERCISE 5 Out of 5000 applicants, only 275 are

chosen. If Hazni is one of the applicants, what is the probability that he is chosen?

A. 11/200B. 200/11C. 189/200D. 200/189

TUTORIALTUTORIAL

GAMING EXAMPLE – 2 DICE

The probability of outcomes of throwing 2 dice! What are the odds of choosing the right numbers - winning!

Two dice totals

Die 1Die 2

1 2 3 4 5 6

1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

P(E) = P(E) = n (E)n (E) n (S) n (S)

The probability of throwing any given total (an EVENT) is the number of ways to throw that total divided by the total number of combinations (SAMPLE SPACE(36)).

Total Number of combinations Probability

2 1 2.78%3 2 5.56%4 3 8.33%5 4 11.11%6 5 13.89%7 6 16.67%8 5 13.89%9 4 11.11%

10 3 8.33%11 2 5.56%12 1 2.78%

Total 36 100%

RulesIf you roll a total of 7 or 11 on the first roll,

you win.If you roll a total of 2, 3, or 12 on the first

roll, you lose.If you roll a total of 4, 5, 6, 8, 9, or 10 on

your first roll, this number becomes your point

Let’s look at the game of STRAIGHT CRAPS?

Two dice totals

Die 1Die 2

1 2 3 4 5 6

1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

what is the probability of winning if you initially roll a total of 5?

???????

GOT the answer yet?

ANSWER!!!!probability of rolling a 5 is 4/36There are ten cells containing 5 or 7 the probability of rolling a 5, and then

rolling a 5 before a 7 is (4/36)(4/10), or about 4.44%.

Two dice totals

Die 1Die 2

1 2 3 4 5 6

1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Here is a summary of your winning probabilities in craps:

INITIAL TOTAL

PROBABILITY OF WINNING PROBABILITY

4 (3/36)*(3/9) 0.027778 5 (4/36)*(4/10) 0.044444 6 (5/36)*(5/11) 0.063131 7 6/36 0.166667 8 (5/36)*(5/11) 0.063131 9 (4/36)*(4/10) 0.044444

10 (3/36)*(3/9) 0.027778 11 2/36 0.055556

TOTAL 0.492929

CONCLUSION

You are making what is known as a Pass Bet. As the table indicates, the probability of winning a Pass Bet is 49.29%.

The casino (sometimes called the house) has a 50.71% probability of winning.

*Hence, the casino has a 1.4% advantage on any Pass Bet.

Conclusion (cont.)

Probability can help you find the advantages and eventually the payoff of playing any casino games!

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POWERS AND POWERS AND EXPONENTSEXPONENTS

LEARNING OUTCOMESTOPICTOPICEXAMPLES

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LEARNING OUTCOMESLEARNING OUTCOMES At the end of the class, students should

be able to:1. Identify index notations2. Define laws of indices3. Simplify expressions using laws of indices4. Solve real life problems

Powers MENUPowers MENU

2) Powers and 2) Powers and ExponentsExponents

Laws of indices

Index notations

Exercises

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Time Value of MoneyTime Value of Money

Powers and Powers and ExponentsExponents

2.1 2.1 Index notations

POWERS topic POWERS topic MENUMENU

2.2 Laws of indices2.2 Laws of indices

LAWS OF INDICES

POWERS topic MENU

2.3 Exercises2.3 Exercises

2.4 Time Value of Money2.4 Time Value of Money

Application of Powers and Exponents in Finance

Time Value of Money The timing of cash outflows and inflows has

important economic consequences. Money that a firm has in its possession today is

more valuable than money in the future because funds can be used to invest to earn a positive return

Time Value of Money: Know this terminology and notation

FV Future Value (1+i)t

Future Value Interest Factor [FVIF]

PV Present Value 1/(1+i)t Present Value Interest Factor [PVIF]

i Rate per period t # of time periods

Question: Why are (1+i) and (1+i)t called interest factors?Start with simple arithmetic problem on interest:How much will $10,000 placed in a bank account paying 5% per year be worth compounded annually?Answer: Principal + Interest $10,000 + $10,000 x .05 = $10,500

2. Factor out the $10,000. 10,000 x (1.05) = $10,5003. This leaves (1.05) as the factor.

So (1+i)t = (1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)… ·(1+i) for “t” times

1. Find the value of $10,000 earning 5% interest per year after two years. Start with the amount after one year and multiply by the factor for each year. [Amount after one year] x (1.05)

= [$10,000 x (1.05)] x (1.05)

= $10,000 x (1.05)2

= $11,025.

Instead of calculating each of the value of money of each year, we can use exponents’ law of multiplication to find factors of a given interest and period for ease of calculation and reference! Thus creating tables for reference!

am X a n = a mn

Future Value Calculations

Find the value of $10,000 in 10 years. The investment earns 8% for four years and then earns 4% for the remaining six years.

FV = $10,000·(1+i1)·(1+i1)·(1+i1)·(1+i1)·(1+i2)·(1+i2)·(1+i2)·(1+i2)·(1+i2)·(1+i2)

Future Value Interest Factors

Future Value Calculations (cont.)

FV = $10,000·(1.08)·(1.08)·(1.08)·(1.08)·(1.04)·(1.04)·(1.04)·(1.04)·(1.04)·(1.04)FV = $10,000 x (1.08)4 x (1.04)6

FV = $17,214.53

AlternativelyFV = $10,000 x 1.3605 x 1.2653FV = $17,214.41 * difference due to rounding

Present Value Calculations

PV = FV÷ [(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)]

PV = FV ÷ (1+i)t

Example: How much do I need to invest at 8% per year, in order to have $10,000 in One year.

Answers:

One year: PV =10,000 ÷ (1.08) = $9,259.26

Two years: = $10,000 ÷ (1.08) ÷ (1.08) OR $10,000 ÷ (1.08)2 = $8,573 Ten years PV = $10,000 ÷ (1.08)10 = $10,000 ÷

2.1589 = $4,632

Present Value Interest Factors

Can you find the PVIF – present Value interest factors? Example: How much do I need to

invest at 5% per year, in order to have $10,000

1. In 2 years2. In 3 years3. In 10 years

Why do we need all these factors???

All cash flows in the future can be added when converted into present values

NPV makes cash flows of different project comparable

Financial managers can use this technique to evaluate which projects to take on!

Example of NPV

If we assume a 10% interest rate,

40.44$1.130$

1.170$

1.1120$100$ 32 NPV

Example 2: Project 1: $100 a year for 4 years, or Project 2: $500 at the end of 4 years.

NPV of Project 1:

NPV of Project 2:

At an interest rate of r = 10%, NPV of project 1 is $316.99. NPV of project 2 is $341.51.

432 1100

rrrrNPV

Conclusion

Powers and Exponents makes Net Present Value calculations for managers much easier!!

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Solving Linear Solving Linear EquationsEquations

LEARNING OUTCOMESLEARNING OUTCOMESTOPICTOPIC

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Linear MENULinear MENU

3) 3) Linear Linear EquationsEquations

Solving equationsSolving equations

PropertiesProperties

Linear MenuLinear Menu

Economic exampleEconomic example

Properties

Solving equations

How do we calculate the height of a certain building if we have limited information on only surrounding buildings?

Say the HSBC building in London?

Economic Example with linear equations

Economic order quantity is the level of inventory that minimizes the total inventory holding costs and ordering costs.

It is one of the oldest classical production scheduling models.

Assumptions:

demand for a product is constant over the year

each new order is delivered in full when the inventory reaches zero

fixed cost charged for each order a holding or storage cost for each unit held

in storage

The ordering cost is constant. The rate of demand is constant The lead time is fixed The purchase price of the item is

constant i.e no discount is available The replenishment is made

instantaneously, the whole batch is delivered at once.

Problem!

How do we determine the optimal number of units of the product to order so that we minimize the total cost associated with the purchase, delivery and storage of the product ?

(EOQ is the quantity to order, so that ordering cost + carrying cost finds its minimum)

Constructing the equation:

Parameters : Total demand Purchase cost Fixed cost to place the order Storage cost for each item per year

Variable Cost

Q = order quantity Q * = optimal order quantity D = annual demand quantity of the product P = purchase cost per unit C = fixed cost per order (not per unit, in addition

to unit cost) H = annual holding cost per unit (also known as

carrying cost or storage cost) (warehouse space, refrigeration, insurance, etc. usually not related to the unit cost)

So the equation becomes:

Total Cost = purchase cost + ordering cost + holding cost

Purchase cost: This is the variable cost of goods: purchase unit price × annual demand quantity.

This is P×D or PD

Ordering cost: This is the cost of placing orders: each order has a fixed cost C, and we need to order D/Q times per year.

This is C × D/Q

Holding cost: the average quantity in stock (between fully replenished and empty) is Q/2.

so this cost is H × Q/2

The equation is :

This looks like what we did before , no?

Now we can find the optimal point by:

To determine the minimum point of the total cost curve, set its derivative equal to zero:

The result of this derivation is:

Solving for Q gives Q* (the optimal order quantity):

Now all you need is to know is C, D and H

If you know that: H = annual holding cost per unit = $500 C = fixed cost per order =$100 D = annual demand quantity of the product =

2000 units

What is the quantity at which a company should order new inventory to minimize cost? REMEMBER:

Answer:

Q* = SQRT (2 *100*2000 /500 ) =???????

ECONOMIC APPLICATIONS

A BRANCH OF SCIENCE CALLED DECISION SCIENCE USES LINEAR EQUATIONS TO SOLVE ECONOMIC PROBLEMS USING OPTOMIZATION / MINIMIZATION!

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MMC Math 2009

Education