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Independent and dependent events and conditional probability

Independent and dependent events and conditional probability

Up to this point, we’ve been focusing on independent events, which are events that don’t effect one another. For example, if I flip a coin two times in a row, the result of the first flip doesn’t effect the second flip, so those flips are independent events.

In other words, if I get heads on the first flip, the second flip still has an equally likely chance of producing heads or tails. If instead I get tails on the first flip, the second flip still has an equally likely chance of producing heads or tails.

Finding volume for triple integrals using spherical coordinates

Finding volume for triple integrals using spherical coordinates

We can use triple integrals and spherical coordinates to solve for the volume of a solid sphere. To convert from rectangular coordinates to spherical coordinates, we use a set of spherical conversion formulas.

All about the sampling distribution of the sample mean

All about the sampling distribution of the sample mean

Consider the fact though that pulling one sample from a population could produce a statistic that isn’t a good estimator of the corresponding population parameter. To correct for this, instead of taking just one sample from the population, we’ll take lots and lots of samples, and create a sampling distribution of the sample mean.

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Discrete probability distributions for discrete random variables

Discrete probability distributions for discrete random variables

A discrete random variable is a variable that can only take on discrete values. For example, if you flip a coin twice, you can only get heads zero times, one time, or two times; you can’t get heads 1.5 times, or 0.31 times.

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Exponential equations to model population growth

Exponential equations to model population growth

The population of a species that grows exponentially over time can be modeled by P(t)=Pe^(kt), where P(t) is the population after time t, P is the original population when t=0, and k is the growth constant.

Surface area of revolution around the x-axis and y-axis

Surface area of revolution around the x-axis and y-axis

We can use integrals to find the surface area of the three-dimensional figure that’s created when we take a function and rotate it around an axis and over a certain interval. The formulas we use to find surface area of revolution are different depending on the form of the original function and the axis of rotation.

$Quotient rule for exponents to simplify fractions$

Quotient rule for exponents to simplify fractions

This is the rule we use when we’re dividing one exponential expression by another exponential expression. The quotient rule tells us that we have to subtract the exponent in the denominator from the exponent in the numerator, but the bases have to be the same.

Solving problems with the formula for distance, rate, and time

Solving problems with the formula for distance, rate, and time

Before you can use the distance, rate, and time formula, D=RT, you need to make sure that your units for the distance and time are the same units as your rate. If they aren’t, you’ll need to change them so you’re working with the same units.

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Finding radius and interval of convergence of a Taylor series

Finding radius and interval of convergence of a Taylor series

Sometimes we’ll be asked for the radius and interval of convergence of a Taylor series. In order to find these things, we’ll first have to find a power series representation for the Taylor series.

Inverse variation of two variables and the constant of variation

Inverse variation of two variables and the constant of variation

The main idea in inverse variation is that as one variable increases the other variable decreases, which means that if x is increasing y is decreasing, and if x is decreasing y is increasing. The number k is a constant so it’s always the same number throughout the inverse variation problem.

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Finding the circumference of a circle

Finding the circumference of a circle

The circumference of a circle is the distance around the circle (its perimeter). You can find the circumference if you know either the radius or the diameter.

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Finding gradient vectors for multivariable functions

Finding gradient vectors for multivariable functions

The gradient vector formula gives a vector-valued function that describes the function’s gradient everywhere. If we want to find the gradient at a particular point, we just evaluate the gradient function at that point.

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Reflecting figures in coordinate space

Reflecting figures in coordinate space

A reflection is a type of transformation that flips a figure over a line. The line is called the line of reflection, or the mirror line, and the line of reflection can be horizontal, vertical, or diagonal.

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Calculating test statistics for means and proportions for one- and two-tailed tests

Calculating test statistics for means and proportions for one- and two-tailed tests

With any hypothesis test, we need to state the null and alternative hypotheses, then determine the level of significance. We’ve already covered these first two steps, and now we want to learn how to calculate the test statistic, which will depend on whether we’re running a two-tail test or a one-tail test.

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Using laws of logarithms (laws of logs) to solve log problems

Using laws of logarithms (laws of logs) to solve log problems

Laws of logarithms (or laws of logs) include product, quotient, and power rules for logarithms, as well as the general rule for logs (and the change of base formula we’ll cover in the next lesson), can all be used together, in any combination, in order to solve log problems.

U-substitution to solve definite integrals

U-substitution to solve definite integrals

U-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. If you don’t change the limits of integration, then you’ll need to back-substitute for the original variable at the end.

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Exponents for simplifying products

Exponents for simplifying products

Exponents are a tool we can use to write numbers in a simpler way. An exponent is a little number that you write above and to the right of another number, and it tells you how many times to multiply the base number by itself.

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Using the comparison test to determine convergence or divergence

Using the comparison test to determine convergence or divergence

The comparison test for convergence lets us determine the convergence or divergence of the given series by comparing it to a similar, but simpler comparison series. We’re usually trying to find a comparison series that’s a geometric or p-series, since it’s very easy to determine the convergence of a geometric or p-series.

Finding unit price (price per unit)

Finding unit price (price per unit)

When we talk about unit price, what we’re really talking about is the “price per unit” of a product (the price per pound of tomatoes or the price per quart of milk). This is the math that helps us compare the prices of things.

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Finding the vector function for the curve of intersection of two surfaces

Finding the vector function for the curve of intersection of two surfaces

When two three-dimensional surfaces intersect each other, the intersection is a curve. We can find the vector equation of that intersection curve using three steps.

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