IM3/IM3H Second Semester Final Exam Study Guide

Comprehensive vocabulary flashcards covering probability, statistics, trigonometry, and financial math based on the IM3/IM3H second semester final exam study guide by Jay Saltzman.

Probability

The chance that something will occur, expressed as $0 \le P \le 1$ or $0\% \le P \le 100\%$.

Independent Events

Two events where the outcome of neither one affects the chances or probability of the other.

Complement of event E ($E'$)

Read as “E prime,” this is the set of all outcomes not included in event E, where $P(E') = 1 - P(E)$.

Pair of Dice Outcomes

A pair of 6-sided perfect cubes with numbers 1-6 has a total of $36$ possible outcomes.

With Replacement

A condition where a first object chosen from a set of objects is replaced before the second object is chosen.

Without Replacement

A condition where the first object chosen from a set of objects is NOT replaced before the second is chosen.

Addition Rule for Union of Events

The formula to avoid double counting overlap: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

Frequency Table

A table of data that shows the frequencies observed or associated with possible outcomes, including “Total” rows and columns.

Conditional Probability ($P(A|B)$)

The probability that event A will occur given that we already know that B is true; calculated as $\frac{P(A \text{ and } B)}{P(B)}$.

Fundamental Counting Principle

If one event occurs in $m$ ways and a second in $n$ ways, the total number of ways they occur in sequence is $m \times n$.

Factorial ($n!$)

A mathematical operation that multiplies a positive integer $n$ by all positive integers less than it down to 1 ($0! = 1$).

Permutation

An ordered arrangement of objects; the number of permutations of $n$ distinct objects taken $r$ at a time is ${}_nP_r = \frac{n!}{(n-r)!}$.

Distinguishable Permutations

The number of distinct arrangements of $n$ objects where groups of objects are of the same type: $\frac{n!}{n_1! \times n_2! \times n_3! \dots n_k!}$.

Combination

The number of ways to choose $r$ objects from $n$ objects without regard to order: ${}_nC_r = \frac{n!}{(n-r)!r!}$.

Mean

The weighted average, found by taking the sum of the data divided by the number of data points.

Median

The middle of the data; it is either the middle number (odd number of data) or the average of the two middle numbers (even number of data).

Mode

The specific data value that appears most frequently in a set.

Range

The full spread of the data, calculated as highest minus lowest.

Variance

The average (mean) of the squared distances of each data point from the mean.

Standard Deviation

The square root of the variance, representing the “average” spread of the data around the mean.

Quartiles ($Q_1, Q_2, Q_3$)

Values that separate data into four parts; they represent the $25^{\text{th}}$, $50^{\text{th}}$, and $75^{\text{th}}$ percentiles respectively.

Normal Distribution

A symmetric “Bell” curve where mean, median, and mode are at the center; describing natural data like heights or IQ scores.

Empirical Rule (Normal Distribution)

Area under the curve: about $34\%$ between $\mu$ and $\mu + \sigma$, $13.5\%$ between $\mu + \sigma$ and $\mu + 2\sigma$, and $2.4\%$ between $\mu + 2\sigma$ and $\mu + 3\sigma$.

Pythagorean Theorem

For right triangles, $a^2 + b^2 = c^2$, where $a$ and $b$ are legs and $c$ is the hypotenuse.

Sine ($\sin(\theta)$)

The trigonometric ratio defined as $\frac{\text{Opposite}}{\text{Hypotenuse}}$.

Cosine ($\cos(\theta)$)

The trigonometric ratio defined as $\frac{\text{Adjacent}}{\text{Hypotenuse}}$.

Tangent ($\tan(\theta)$)

The trigonometric ratio defined as $\frac{\text{Opposite}}{\text{Adjacent}}$ or $\frac{\sin(\theta)}{\cos(\theta)}$.

Cosecant ($CSC$)

The reciprocal of Sine, defined as $\frac{\text{Hypotenuse}}{\text{Opposite}}$.

Secant ($SEC$)

The reciprocal of Cosine, defined as $\frac{\text{Hypotenuse}}{\text{Adjacent}}$.

Cotangent ($COT$)

The reciprocal of Tangent, defined as $\frac{\text{Adjacent}}{\text{Opposite}}$ or $\frac{\cos(\theta)}{\sin(\theta)}$.

Triangle Inequality

Principle stating any two side lengths of a triangle must add up to more than the third side: $|a - b| < c < a + b$.

Law of Sines

The ratio of each side to the sine of its opposite angle is constant: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

Law of Cosines

A generalized version of the Pythagorean Theorem for all triangles: $c^2 = a^2 + b^2 - 2ab \times \cos(C)$.

Human Capital

Skills, education, and experience that increase a person's productivity.

Rate of Return

The gain or loss on an investment over time, expressed as a percentage.

Capital Gains

Profits earned from selling assets that have increased in value.

Mutual Fund / ETF

A pooled investment that holds many different stocks or assets simultaneously.

Roth IRA

A retirement account that offers growth with no capital gains tax.

Dollar-Cost Averaging

Investing a fixed amount regularly regardless of market conditions.

Diversification

Spreading investments across different asset types, industries, and regions to reduce risk.

Compound Interest

Earning interest on both the original principal and on prior earned interest.

Emergency Fund

$3$–$6$ months of living expenses set aside in liquid form to cover unexpected costs.

Credit Score

A numerical score ranging from $300$–$850$ summarizing an individual's ability to get credit.

Insurance Premium

The regular amount paid to maintain an insurance policy.

Insurance Deductible

The amount one must pay out-of-pocket before insurance covers the rest of a claim.

Exponential

$$a^{n}=b$$

Base=a, exponent=n, value=b

Logarithm

$$^{Log}a^{b=n}$$

Base=a Value=b Exponent=n

Product Rule

$$Log_{a}\left(xy\right)=log_{a}x-\log_{a}y$$

Quotient Rule

$$Log_{a}\left(\frac{x}{y}\right)=\log_{a}x-\log_{a}y$$

Power Change Rule

$Log_{a}x^{p}=p\cdot\log_{a}x$