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Formulae and Equations

Properties of Probability

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Overview:

Probability is the measure of the likelihood of an event.  The basic mathematics of probability theory started with games of chance, but it can be applied to many situations, from weather forecasting to politics.  Probabilities range from 0 (no likelihood) to 1 (certainty), and are expressed as rational numbers.

What Is the Sample Space?

The sample space is the set of all possible outcomes of an event.  For a coin toss, the coin will either result in heads or tails.  For the roll of one die, the sample space is all the values on the faces of the die, or a set of {1, 2, 3, 4, 5, 6}.  For the roll of a pair of dice, the sums will be in a set from 2 (both dice give you a 1), the smallest sum possible, to 12 (both dice give you 6), the largest sum possible.

What Is A Fair Experiment?

In a fair experiment, all possible outcomes are equally likely.  The probability of any outcome is related to the total number of outcomes by a ratio of the number of outcomes in that event to the number of all possible outcomes of the event (the sample space).  Therefore, the probability that a coin will be heads is 1/2.  The probability that if one die is rolled, the number on top will be a 3 is 1/6.

What If Events Are Not Equally Likely?

Sometimes, possible outcomes can be combined in such a way so that not all outcomes are equally likely.  Suppose two fair coins are tossed: there are 4 possibilities in the sample space {HH, HT, TH, TT}.  The probability of each event when order is important equals 1/4 for each possibility. However, if the question is merely “How many heads come up when two coins are tossed?”, there are only 3 possibilities in the sample space, 0 heads, 1 head, or 2 heads.  The event 0 heads is defined as {TT}, and the event 2 heads is defined as {HH}.  However, there are two possibilities for 1 head, either {HT} or {TH}, so the outcomes are not equally likely.

What Are Mutually Exclusive Events?

If events are mutually exclusive, it means that neither sample space A or sample space B contain common elements.  Therefore, the probabilities can be added to form the probability of one event or the other occurring.  The sample space for tossing two fair dice and getting a sum of 7 consists of {(6,1), (1, 6), (2, 5), (5,2), (3,4), and (4,3)}.  The sample space for tossing two fair dice and getting a sum of 11 consists of {(6,5), (5,6)}.  The sample space for getting either a sum of 7 or a sum of 11 is the union of both sets.

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Math Review of Random Numbers

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Overview:  What Are Random Numbers?

In a set of random numbers, the numbers do not follow any pattern.  Each number has an equal probability of occurring, and each number event is independent of any others .  Most of the time, numbers that are close to random are generated by computer programs or calculator programs designed to do just that.

What Are Some Examples of Random Events?

Many events are close to random. For example, individual molecules within a gas tend to move randomly, so that it cannot be predicted where an individual molecule will be.  Similarly,  the theory of radioactivity predicts that a percentage of atoms in a substance will decay into isotopes given an amount of time, but it does not predict precisely which particular atom will decay.

How Are Random Numbers Found?

Numbers that are close to random can be found by consulting random number tables, as the result of computer programs to generate random numbers, and by using a calculator to generate random numbers.  In addition, games that depend on giving all the players a fair chance are often determined by using dice or a spinner divided into equal parts.  That way each number has an equal probability of being chosen.

How Is Randomness Used In Statistics?

In scientific experiments, statistics are used as a tool to judge the results of an experimental treatment.  Subjects have an equal chance to be assigned to a treatment condition through random assignment.  Often a random number generator is used in order to assign subjects to treatment groups.  Not only does each subject have an equal chance of being assigned to any of the treatment groups, but the choice of any one subject is independent of all the others.  This minimizes errors that could occur if the assignment is not random, but confounded.

What Are Monte Carlo Methods?

Monte Carlo methods are mathematical simulations that use random numbers to generate solutions.  They are called Monte Carlo methods because the earliest studies used equipment to generate random numbers similar to the methods in gambling casinos, such as decks of cards and roulette wheels.  Some simulations include the common practice of airlines overbooking seats on flights, because there are probabilities generated for how many passengers will arrive to take seats.  The computer simulation can be run for the number of seats on the airplane as the number of trials, using random numbers to suggest a possible outcome.

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Stoichiometry: Dealing with Excess and Limiting Reactants

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Overview:

In many chemical reactions there is an excess reactant a limiting reactant. The amount of product produced is determined by the stoichiometric calculations using the amount of limiting reactant present in the reaction. This means that not all of the excess reactant is used up during the reaction since there is no more of the other reactant present to react with.

Simpler Terms:

It is easier to think of limiting and excess reactants in terms of baking. If a recipe calls for 1 cup of peanut butter, and 3 cups of sugar to make 12 cookies and you start with 3 cups of peanut butter and 12 cups of sugar based on the amount of peanut butter you have you could make 36 cookies but based on the amount of sugar you have you could make 48 cookies. In this case peanut butter is our limiting reactant and we can only make 36 cookies but will have 3 cups of sugar left over when we are finished which means it is our excess reactant.

The idea behind limiting and excess reagents is identical to what we did above with the baking except we are dealing with elements instead of ingredients.

Example Question:

Four moles of propane reacts with ten moles of oxygen. How much carbon dioxide will be produced from this reaction?

Step One:

The first step in determining the excess and limiting reactants is to have your equation fully balanced so you can see the stoichiometric ratios between the different compounds.

C3H8 + 5 O2 → 3 CO2 + 4 H2O

Step Two:

The next step is to determine the conversion factor to multiply the moles of each element by to find the amount of product it will produce. To find the conversion factor simply divide the element by its coefficient and then multiply by the coefficient of the element you wish to know the amount of moles of.

O2 → CO2 conversion factor = 3/5

C3H8 → CO2 conversion factor = 3/1

Step Three:

Repeat step two for all the reactants in the equation with the amount of moles you have for that reactant and whichever one yields the least amount of product is the limiting reactant. All the other reactants are excess reactants. Use the limiting reactant for the amount of product formed.

Ten moles of O2 will produce, 10 * 3/5 = 6 moles of CO2 produced

Four moles of C3H8 will produce, 4 * 3/1 = 12 moles of CO2 produced

To answer the question at the beginning, when four moles of propane reacts with ten moles of oxygen only 6 moles of carbon dioxide will be produced and there will be an excess of propane at the end.

How to Calculate Work Done by a Force

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Work is the energy applied to an object as it moves some distance. The amount of work done is directly proportional to the magnitude of force applied, as well as the displacement of the object. In some cases, there may be an angle between the direction of displacement and force vector.

The force must be perpendicular to the direction of displacement in order to produce work. This can be considered through application of trigonometry, where the angle is found between the displacement distance and force vector. When the force opposes the direction of displacement, the work produced is negative.

Work Done by a Force 2

Work is a scalar quantity as it does not have a defined direction. The unit for work is the Newton meter (Nm), since force is measured in Newtons (N) and displacement is measured in meters (m). Most commonly, work is written in units of joules (J), an SI unit. All types of energy, such as heat and potential energy, are measured in terms of joules.

Work Done by a Force 3

We can determine the work done by specific forces in varying scenarios. As a hockey puck slides across a surface, multiple forces are present and play separate roles. No work is done by the normal force or gravitational force since the vectors are perpendicular to the direction of displacement. On the other hand, the friction force is parallel to the surface, and does work on the hockey puck. However, the friction force produces negative work. This will always be the case as friction constantly opposes movement.

Work Done by a Force 4

In the diagram above, an 8 kg crate is dragged 10 m across a wide room. The pulling force was applied with 20 N at a 30°angle. The coefficient of kinetic friction between the crate and carpeted floor is 0.1. What is the work done by each of the forces applied on the crate?

Work Done by a Force 7

 

A free body diagram can be produced to show all forces acting on the crate. The applied force can be broken down to the x-component and y-component through trigonometry. Because they are parallel to the direction of movement, the x-component of the applied force and the friction force both produce work. The remaining forces are perpendicular and do not produce work.

Work done by FX : WFX = F ∙ d ∙ cosθWFX = 20 ∙ 10 ∙ cos30°WFX = 173.21 J
Work done by Ff : WFf = Ff ∙ d ∙ cosω

WFf = μk ∙ FN ∙ d ∙ cosω

WFf = μk ∙ (FG – FY) ∙ d ∙ cosω

WFf = μk ∙ (mg – Fsinθ) ∙ d ∙ cosω

WFf = 0.1 ∙ ((8 ∙ 9.81) – 20sin30°) ∙ 10 ∙ cos180°

WFf = – 68.48 J

The basic formula for work can be applied to both forces, although some forces may have to be broken down to simpler terms. The downward and upward forces are in equilibrium, and as a result, the normal force can be determined. The difference between the gravitational force and y-component of the applied force is equivalent to the normal force. The total work of the system can be determined through the sum of work values.

Wtotal = WFX + WFf

Wtotal = 173.21 J – 68.48 J

Wtotal = 104.73 J

Dimensional Analysis

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Dimensional Analysis can be very useful when you come across a situation where you forget a formula that applies to certain values. This method is also proven to be beneficial when converting between units within a problem.

For example, you are asked to find the time it took for a car to travel 200 meters with a speed of 10 meters per second. You know that you want to end up with a unit of time, given units of meters per second and meters. We want to somehow get rid of the meters to end up with units of seconds. To get this, we will simply divide meters by meters per second. If that was too confusing, you can easily visualize it this way:

Going back to the problem, we will simply do:

Dimensional analysis is also very useful in consolidating multiple conversion steps into one.

For example, we want to convert 85 miles per hour into SI units, or meters per second.

We know the following information:

1 mile = 1609.34 meters or 1609 meters when rounded to the nearest ones digit
1 hour = 60 minutes
1 minute = 60 seconds

We can then set up the same ratios as the example from above, and we can write what we know in the following way:

What would have taken several more steps to do, took only one step with this approach. This is a great method to check that you are using the right formula because it allows you to make sure that your units are correct (ie. you are looking for an answer with units of length, but through careless calculations you end up with seconds). You can use this method to make sure that you are calculating the right value as well as converting between your units in an organized manner.

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This article was written for you by Frances, one of the tutors with TestPrep Academy.

From Standard to Vertex and Back!

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Parabolic equations can be some of the trickiest types of equations out there. One of the most common problems students have when dealing with parabolas is when it is given in a form they don’t recognize. If you can’t understand the equation, you can’t use it properly. The two forms I will discuss are known as standard and vertex forms. Standard form is in the form of a standard quadratic equation (ie. ax^2 + bx + c) where as the vertex form is just a manipulation of the same form.

Vertex to Standard

To get an equation from vertex form to standard form you simply expand and simplify what is bracketed, known as the binomial square.

Example: Let’s use the equation
y=(x-1)^2 + 1
We start off by expanding
(x-1)^2
We are then left with
y=x^2 – 2x + 1 + 1
Finally we simplify to get our final answer in standard form:
y = x^2 – 2x + 2

Standard to Vertex

To convert to vertex form when given a equation in standard form you need to understand how to complete the square. For more experienced students, they man have no trouble with this, but for those in need of a little help there is a general formula:

(x + t/2)^2 = x^2 + 2(t/2)x + (t/2)^2

Example: (x + 3)^2 = x^2 + 2(3)x + (3)^2 = x^2 + 6x + 9

It won’t always be as simple as just completing the square. Often times you will need to add or subtract a x0 term, or a term with no variable.

Example: Let’s use the equation
y = x^2 + 8x + 13
When we first look at it, we can see that this equation does not satisfy the general formula for completing the square. In order to complete the square we must add 3to both sides.
3 + y = x^2 + 8x + 16
We then rearrange in order to isolate y on the left side.
Y = x^2 + 8x + 16 – 3
We can now complete the square using the first three terms on the right hand side.
Y = (x + 4)^2 – 3
This is now in vertex form!

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This article was written for you by Troy, one of the tutors with Test Prep Academy.

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