Answer to probability question Part 2

I was asked if the chances of flipping two same colour cards at the same time would be equal to 1/3. The answer is yes. There are 4c2 (6) number of possible combinations for flipping 2 cards. Only two of the six combinations are made up of cards of the same number. Therefore it is true that the chance of flipping two cards of the same colour is equal to 2/6 or 1/3.

Answer to probability question

Ron and Kenneth decide to gamble. Four aces are taken from an ordinary deck of cards. The four aces are placed faced down and Kenneth is to flip 2 over cards. If both aces are of the same colour Kenneth wins 50 dollars. If not Ron will win 50 dollars.
Kenneth disagrees to this wager and requested that he wins 100 if he flips 2 of the same colour.

Is this a fair bet?

The answer is yes.
You may be wondering why it is fair for Kenneth to win 100 whereas Ron only gets 50. If we look at all the possible outcomes of flipping cards we get four possibilities. RR, RB, BR and BB. It may seem like there is a 50-50 chance of flipping the same colour but this is incorrect.

If we take another approach, if the first flipped card was red, there would be one red in the 3 unflipped cards and there would be a 1/3 chance of flipping that red. This is the same case as if the first card flipped was black. Therefore the chance of flipping two cards of the same colour is actually 1 in 3 not 50-50. Therefore it is fair for Kenneth to win double because he is half as likely to win.

Probability Problem

Ron and Kenneth decide to gamble. Four aces are taken from an ordinary deck of cards. The four aces are placed faced down and Kenneth is to flip 2 over cards. If both aces are of the same colour Kenneth wins 50 dollars. If not Ron will win 50 dollars.
Kenneth disagrees to this wager and requested that he wins 100 if he flips 2 of the same colour.

Is this a fair bet?

Polynomials

Polynomials is a harder topic because it requires the basic knowledge before the questions can be completed. Once you are used to the questions, the topic becomes rather simple.

Polynomials are expressions created by the addition, subtraction or multiplication of variables.
Eg, x^3+ 2x^2 + 3 is a polynomial
x^1/2 + 2 is not a polynomial because it contains a rooted variable
x^-1 + 3x is not a polynomial because it contains a negative power
2^x + 5 is not a polynomial because it contains a variable as a exponent
^ is to the power of.

Polynomials can only have one variable and the powers of that variable must be positive integers.

Terminology
Roots – Also called zeroes, are the x intercepts or solutions of the polynomial.
Leading Term – The term in which the variable has the highest power
Leading Coefficient – The Coefficient of the leading term
Monic – A polynomial is monic of the leading coefficient is equal to 1

Dividing and multiplying polynomials
To divide polynomials we use the classic long division method.
To multiply polynomials it is exactly the same as expanding the brackets except there may be many more terms.

Demonstration of division and multiplication of polynomials

Factor Theorem
A factor of a polynomial is a smaller polynomial that can be divided into the original polynomials and leave a 0 remainder. If a is a root of the polynomial then (x-a) is a factor of the polynomial. Similarly if P(a) = 0 then when (x-a) is a factor.

Remainder Theorem
If a polynomial is divided by (x-a) then the remainder is P(a)

For those who are unfamiliar with the concepts of functions
Basically if P(x) = x^3 + x^2
P(1) = 1^3 + 1^2
P(-2) = (-2)^3 + (-2)^2

Sketching polynomials
A polynomial can be easily sketched if it is in factored form. Such as (x-1)(x-2) or (x+3)(x-1) (x-4). Basically to sketch we first determine all the roots. If it is a positive polynomials then start from the top right quadrant, if it is negative then start from the bottom right quadrant. Draw a curved line through all the roots and that will basically be the graph.

Repeated roots
If (x-1) is a factor of a polynomial and it is also a factor of the resulting polynomial, then (x-1) is a double root. A even root touches the x axis in the shape similar to a parabola. A odd repeated root cuts the x axis in a shape similar to a y=x^3 (cubic graph).

Finding Factors
To find factors of a polynomial it is basically a process of trial and error. All the roots of a polynomial are factors of the constant term. This is very useful information as it simplifies the trial and error process.

Anyways this should be everything that is covered in Year 10 Polynomials.
Bye

Circle Geometry Proofs

Logarithms

Logarithms are basically the opposite of exponentials. By using the logarithm function, we can find the value of the variable if it is an exponent.

Click on the images for a larger version.

Last question of the Level 3 test

Now that everyone should have finished their Level 3 test, I am going to post the solutions to the last question.

Question: A number of students are going on an excursion that is of a set price. 2 students decided that they would not like to participate and the rest of the students had to pay an extra 36cents. Before the excursion 1 of the 2 students decided to join and the other students were refunded 20cents. How many students are going on the excursion.

Common Question that I got asked many times

Good Luck everyone for maths

Analytic Geometry

Analytic geometry, also know as coordinate geometry is the study of geometry using a coordinate system. This topic is very important because the knowledge is required for topics in Year 11 and 12 mathematics such the Parabola as a Locus and Conics. Anyways the following pictures basically sum up what is required for Year 10.

Click on the images if you would like to see an enlarged version, anyways, the proofs are not necessary. I only put them in just in case you would like to know where the formulas came from. If there is anything unclear, please ask in the cbox or on msn. This blog need more exposure, so please spread the word.

Solution to the question

Mail Merge

I'm writing this post to help people with mail merging which will be required tomorrow.

Create an Excel Spreadsheet with the information you need. Save it in a location that can be easily accessed such as the desktop.

Click on select recipients and choose use existing file.

Locate your file.

Click on insert merge fields and choose the fields that you want to include.

Click on preview fields to look through your different mailing addresses.

An enlarged version of the image can be viewed if you click on it.

Try to find a and b

Given that i^2 = -1 find values for a and b.

Solution can be found here.

A Picture That Makes Me Happy

Well this picture doesn't really make me happy but I like it because of the maths joke. Anyways most people would know that pi is an irrational number. As for i, it is known as the imaginary unit where 1^2 = -1. You should know that -1 cannot be square rooted with with advanced mathematics it can be.

Thanks to Raymond G for linking me to this image.

Trigonometry

Trigonometry is the study of triangles and the relations between their sides and angles. There are three main trigonometric functions namely: Sine, Cosine and Tangent (Sin, Cos, Tan).
In a right angled triangle, Sin = Opposite/Hypotenuse Cos= Adjacent/Hypotenuse Tan= Opposite/Adjacent
The easy way to remember this is by remembering SOHCAHTOA.

These ratios can be extended in order for it to work out unknown side or angles of non-right angled triangles.

The area of a triangle can also be determined by using trigonometry.

Proofs for the Sin and Cos rules and the Area of triangle formula can be found here.

Unit Circle
The unit circle can be used to find the the Sin, Cos and Tan of angles greater than 90 degrees. A unit circle is a circle where the radius is equal to 1 unit.

Exact Values

The easy way to remember these exact values is that Sin 0, 30, 45, 60 and 90 are √0/2, √1/2, √2/2, √3/2 and √4/2 (1) respectively. The same goes for Cos except the order is reversed. To work out Tan, you simply have to divide Sin by Cos. Click on the images to view an enlarged version.

What's Up Doc

This post will be a review of the physics topic hoping to aid you in the test tomorrow.

Newton's three laws of motion:

1. Law of Inertia: An object will stay at rest or move in a straight line with a constant velocity unless acted upon by an external force.
2. F=ma An object of a given mass will accelerate at a rate proportionate to the force.
3. For every action, there is an equal opposite reaction.

Speed vs. Velocity

• Speed is scalar (size only)
• Speed = Distance/Time
• Velocity is a vector (size and direction)
• Velocity = Displacement/Time

Displacement is the shortest distance between the starting point and the current point while distance is the total path traveled.

Acceleration is the rate the velocity changes with respect to time.
Formula for average acceleration is (difference in velocity)/time.

Equations of motion:

Where v is the final velocity, u is the initial velocity, a is acceleration and t is time

Where r is displacement.

Newton's Universal Gravitational Law
The force between two bodies can be calculated by using the following formula

Where m1 and m2 are the masses of the two bodies and r is the distance between the centre of the two bodies. G is the Universal gravitational constant.

The gravity of a body can be calculated by using the following formula

Centripetal Force
In centripetal motion, the acceleration is equal to the velocity squared divided by the radius.
Therefore F=ma become F=m(v^2/r)

Calculating Average Acceleration using a ticker timer.
A ticker timer will make 50 marks per second in other words, the time between the dots is equivalent to 1/50 of a second. To find the average acceleration you would have to find the initial velocity (distance between the first two dots/(1/50)) and the final velocity (distance between last to dots/(1/50)). Find the difference between the two velocities and then divide that by the total time taken.

Other notes:
The pronumeral for distance is 's' not 'd'
Displacement is 'r' not 'd'
Speed/Velocity is 'v'not 's'

Good luck for the test tomorrow

Simultaneous Equations

I am writing this post because I've seen several problems with the solving of simultaneous equations.
Eg. If xy = 6 and x+y = 5 find x and y
      x = 5-y
      5y-y^2 = 6
      (y-2)(y-3)=0
      y= 2 or 3
If y = 2, x=3 and if y = 3, x =2

If you are trying to solve a simultaneous equation with a function and a non-function, then final substitution must be made into the function otherwise it may yield incorrect results.
Eg. Solve simultaneously
      x^2 + y^2 = 16 (circle graph - not a function)
      3x - 4y - 20 = 0
      y = (3x-20)/4
Substituting into the first equation gives you
x^2 + [(3x-20)/4]^2 = 16
x^2 + (9x^2 - 120x + 400)/16 = 16
16x^2 + 9x^2 - 120x + 400 = 256
25x^2 - 120x + 144 = 0
(5x - 12)^2 = 0
x = 12/5 At this point, x must be substituted into the function to find the y value, otherwise it will give more than one result because there are two y values for just about every x value.

Another note sqrt(x^2 + y^2) ≠ x+y
(x + y)^2 ≠ x^2 + y^2

Introductory Calculus

Calculus is a major part of mathematics that studies change. There are two major branches, namely Differential Calculus and Integral Calculus which are linked together by the Fundamental Theorem of Calculus.

Differential Calculus
Differential Calculus is the study of the rates at which a function changes. In a straight line graph, this would be the gradient, but in curves, it would be the gradient of the tangent at particular points. We find the gradient of these tangents through a process called differentiation. The derivative of a function (function after differentiation) can be written as

There are a few rules for differentiating:

1. Power Rule
   
   
   
   
   
   
   
   
   
   
   
   
2. Product Rule: The y is a product of two functions, then the derivative is:
   
   
   
   
   
   
   
   
3. Quotient Rule: if y is a function divided by another function, then the derivative of y is:
   
   
   
   
4. Chain Rule/Function of a Function:
   
   
   
   
   
   
   
   
   
   
   
   
   
5. Differentiation by first principles: The original but long method
   

Integral Calculus
Integral Calculus is the study of areas under functions. In a straight line graph, this would be a trapezium, but in curves it can be very hard to find the exact value, integral calculus can give the exact value. There are two different types of Integrals (Function after integrating). They are the Indefinite (also the antiderivative, opposite of derivative) and Definite (which give the area between the curve and the x axis within the boundaries).

The integral of a function is:

Hope you have learnt something!