Measurement, Counting Methods, and Probability — Study Guide for Liberal Arts Math

Measurement

Metric Units and Conversions

The metric system is a decimal-based system of measurement used worldwide. Understanding metric units and their conversions is essential for solving real-world problems involving length, mass, and capacity.

• Common Metric Units:

  • Length: kilometer (km), meter (m), centimeter (cm), millimeter (mm)

  • Mass: kilogram (kg), gram (g)

  • Capacity: liter (L), milliliter (mL)

• Metric Prefixes:

  • kilo-: 1,000 base units (e.g., 1 km = 1,000 m)

  • centi-: 1/100 of a base unit (e.g., 1 m = 100 cm)

  • milli-: 1/1,000 of a base unit (e.g., 1 L = 1,000 mL)

• Conversion Rule: To convert to a smaller unit, multiply; to convert to a larger unit, divide.

• Examples:

  • 3 meters to centimeters: $3 \times 100 = 300$ cm

  • 5,000 meters to kilometers: $5,000 \div 1,000 = 5$ km

  • 750 milliliters to liters: $750 \div 1,000 = 0.75$ L

Measuring Area and Volume

Area measures the surface inside a two-dimensional shape, while volume measures the space inside a three-dimensional object. Each shape has a specific formula for calculation.

Shape	Formula
Rectangle	$A = l \times w$
Square	$A = s^2$
Triangle	$A = \frac{1}{2} \times b \times h$
Circle	$A = \pi r^2$
Rectangular Prism (Box)	$V = l \times w \times h$
Cube	$V = s^3$

• Variables: $l$ = length; $w$ = width; $b$ = base; $h$ = height; $r$ = radius; $s$ = side.

• Unit Reminder: Area uses square units (e.g., cm2, in2); volume uses cubic units (e.g., cm3, in3).

• Examples:

  • Rectangle: $l = 8$ in, $w = 4$ in; $A = 8 \times 4 = 32$ in2

  • Box: $l = 5$ in, $w = 3$ in, $h = 2$ in; $V = 5 \times 3 \times 2 = 30$ in3

Measuring Weight and Temperature

Weight and temperature are measured using specific units and conversion formulas. Understanding these conversions is important for interpreting scientific and everyday data.

Conversion	Equivalent
1 kilogram	1,000 grams
1 gram	1,000 milligrams
1 pound	16 ounces
1 U.S. ton	2,000 pounds

• Temperature Formulas:

  • Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$

  • Fahrenheit to Celsius: $C = \frac{5}{9}(F - 32)$

• Examples:

  • Convert 20°C to Fahrenheit: $F = \frac{9}{5} \times 20 + 32 = 68$°F

  • Convert 68°F to Celsius: $C = \frac{5}{9}(68 - 32) = 20$°C

Counting Methods and Probability Theory

The Fundamental Counting Principle

This principle states that if a task consists of several independent steps, the total number of outcomes is the product of the number of choices for each step.

• Example: 3 shirts, 2 pairs of pants, 4 pairs of shoes: $3 \times 2 \times 4 = 24$ outfits.

• Use: When a problem asks for the number of possible outcomes from multiple sets of choices.

Permutations

A permutation is an arrangement of items where order matters. Use permutations when the position or sequence is important.

• Formula: $P(n, r) = \frac{n!}{(n - r)!}$

• Factorial: $n! = n \times (n-1) \times \ldots \times 1$

• Example: 6 students compete for 3 places: $P(6, 3) = \frac{6!}{3!} = 120$ arrangements.

• Clue: If switching the order changes the result, use a permutation.

Combinations

A combination is a selection of items where order does not matter. Use combinations when simply choosing a group.

• Formula: $C(n, r) = \frac{n!}{r!(n - r)!}$

• Example: Choose 3 students from 5: $C(5, 3) = \frac{5!}{3!2!} = 10$ groups.

• Comparison:

  • Permutations — order matters (e.g., race placements, arranging people in seats)

  • Combinations — order does not matter (e.g., choosing a committee, selecting a team)

Fundamentals of Probability

Probability measures the likelihood of an event occurring. It is always between 0 (impossible) and 1 (certain).

• Formula: $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}}$

• Example: Probability of rolling a 4 on a fair die: $P = \frac{1}{6}$

• Probability can be expressed as: fraction, decimal, or percentage.

Probability with Counting Methods

Counting methods help determine the total and favorable outcomes before calculating probability.

• Example: Bag with 3 red and 2 blue marbles: $P(\text{red}) = \frac{3}{5} = 60\%$

• Steps: Count possible outcomes, then divide favorable by total outcomes.

NOT, OR, and Odds

These concepts expand basic probability to include complements, unions, and comparisons of favorable to unfavorable outcomes.

• NOT (Complement): $P(\text{not }A) = 1 - P(A)$

• Example: Chance of rain is 30%; chance it will not rain is $1 - 0.30 = 0.70$ or 70%.

• OR (Union): $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Mutually Exclusive Events: If A and B cannot happen together, $P(A \text{ or } B) = P(A) + P(B)$

• Example: Roll a 2 or 5 on a die: $\frac{1}{6} + \frac{1}{6} = \frac{1}{3}$

• Odds: Odds in favor = favorable : unfavorable; odds against = unfavorable : favorable

• Example: Bag with 3 red and 2 blue marbles: odds in favor of red = 3:2; odds against red = 2:3

AND and Conditional Probability

AND refers to the probability of two events both occurring. Conditional probability is the probability of one event given another has occurred.

• AND (Independent Events): $P(A \text{ and } B) = P(A) \times P(B)$

• Example: Flip a coin twice: $P(\text{heads both times}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

• Conditional Probability: $P(A | B) = \frac{P(A \text{ and } B)}{P(B)}$

• Example: 6 girls in a class, 4 wear glasses: $P(\text{glasses} | \text{girl selected}) = \frac{4}{6} = \frac{2}{3}$

• Word Clues: AND = multiplication; OR = addition; NOT = subtract from 1; GIVEN = conditional probability.

Expected Value

Expected value is the average outcome over many trials. It is calculated by multiplying each possible outcome by its probability and summing the results.

• Formula: $E(x) = \sum [x \times P(x)]$

• Example: 50% chance to win $10, 50% chance to lose $4:$E(x) = 10 \times 0.5 + (-4) \times 0.5 = 5 - 2 = 3$

• Interpretation: Positive expected value = average gain; negative expected value = average loss.

Quick Formula Cheat Sheet

Topic	Formula / Rule
Rectangle area	$A = l \times w$
Triangle area	$A = \frac{1}{2} \times b \times h$
Circle area	$A = \pi r^2$
Rectangular prism volume	$V = l \times w \times h$
Celsius to Fahrenheit	$F = \frac{9}{5}C + 32$
Fahrenheit to Celsius	$C = \frac{5}{9}(F - 32)$
Permutations	$P(n, r) = \frac{n!}{(n - r)!}$
Combinations	$C(n, r) = \frac{n!}{r!(n - r)!}$
Basic probability	Favorable outcomes / total outcomes
Complement / NOT	$P(\text{not }A) = 1 - P(A)$
OR	$P(A) + P(B) - P(A \text{ and } B)$
Independent AND	$P(A) \times P(B)$
Conditional probability	$P(A | B) = \frac{P(A \text{ and } B)}{P(B)}$
Expected value	$E(x) = \sum [x \times P(x)]$

Practice and Test-Taking Tips

• Focus on metric conversions, area vs. volume, and temperature conversions.

• Recognize when to use permutations vs. combinations.

• Apply probability concepts involving AND, OR, NOT, and conditional probability.

• Understand factorials and expected value calculations.

• Tip: Before calculating, underline what the question is asking: conversion, area/volume, arrangement, selection, or probability.