Example What is the probability of getting an even number or a multiple of 3 on the roll of a die? The probability of getting an even number on the roll of a die is ᎏ 3 6 ᎏ , because there are three even numbers (2, 4, 6) on a die and a total of 6 possible outcomes. The probability of getting a multiple of 3 is ᎏ 2 6 ᎏ , because there are 2 multiples of three (3, 6) on a die. But because the outcome of rolling a 6 on the die is an overlap of both events, we must subtract ᎏ 1 6 ᎏ from the result so we don’t count it twice. P(even or multiple of 3) = P(even) + P(multiple of 3) – P(overlap) = ᎏ 3 6 ᎏ + ᎏ 2 6 ᎏ – ᎏ 1 6 ᎏ = ᎏ 4 6 ᎏ COMPOUND PROBABILITY A compound event is performing two or more simple events in succession. Drawing two cards from a deck, rolling three dice, flipping five coins, having four babies, are all examples of compound events. This can be done “with replacement”(probabilities do not change for each event) or “without replacement” (probabilities change for each event). The probability of event A followed by event B occurring is P(A) ϫ P(B). This is called the counting principle for probability. Note: In mathematics, the word and usually signifies addition. In probability, however, and signifies multi- plication and or signifies addition. Example You have a jar filled with 3 red marbles, 5 green marbles, and 2 blue marbles. What is the probability of getting a red marble followed by a blue marble, with replacement? “With replacement” in this case means that you will draw a marble, note its color, and then replace it back into the jar. This means that the probability of drawing a red marble does not change from one simple event to the next. Note that there are a total of 10 marbles in the jar, so the total number of outcomes is 10. P(red) = ᎏ 1 3 0 ᎏ and P(blue) = ᎏ 1 2 0 ᎏ so P(red followed by blue) is ᎏ 1 3 0 ᎏ ϫ ᎏ 1 2 0 ᎏ = ᎏ 1 6 00 ᎏ . If the problem was changed to say “without replacement,” that would mean you are drawing a marble, not- ing its color, but not returning it to the jar. This means that for the second event, you no longer have a total num- ber of 10 outcomes, you only have 9 because you have taken one red marble out of the jar. In this case, P(red) = ᎏ 1 3 0 ᎏ and P(blue) = ᎏ 2 9 ᎏ so P(red followed by blue) is ᎏ 1 3 0 ᎏ ϫ ᎏ 2 9 ᎏ = ᎏ 9 6 0 ᎏ . – THEA MATH REVIEW– 110 Statistics Statistics is the field of mathematics that deals with describing sets of data. Often, we want to understand trends in data by looking at where the center of the data lies. There are a number of ways to find the center of a set of data. MEAN When we talk about average, we usually are referring to the arithmetic mean (usually just called the mean). To find the mean of a set of numbers, add all of the numbers together and divide by the quantity of numbers in the set. Average = (sum of set) ÷ (quantity of set) Example Find the average of 9, 4, 7, 6, and 4. ᎏ 9+4+7 5 +6+4 ᎏ = ᎏ 3 5 0 ᎏ = 6 The mean, or average, of the set is 6. (Divide by 5 because there are 5 numbers in the set.) MEDIAN Another center of data is the median. It is literally the “center” number if you arrange all the data in ascending or descending order. To find the median of a set of numbers, arrange the numbers in ascending or descending order and find the middle value. ■ If the set contains an odd number of elements, then simply choose the middle value. Example Find the median of the number set: 1, 5, 4, 7, 2. First arrange the set in order—1, 2, 4, 5, 7—and then find the middle value. Since there are 5 values, the middle value is the third one: 4. The median is 4. ■ If the set contains an even number of elements, simply average the two middle values. Example Find the median of the number set: 1, 6, 3, 7, 2, 8. First arrange the set in order—1, 2, 3, 6, 7, 8—and then find the middle values, 3 and 6. Find the average of the numbers 3 and 6: ᎏ 3+ 2 6 ᎏ = ᎏ 9 2 ᎏ = 4.5. The median is 4.5. – THEA MATH REVIEW– 111 MODE Sometimes when we want to know the average, we just want to know what occurs most often. The mode of a set of numbers is the number that appears the greatest number of times. Example For the number set 1, 2, 5, 9, 4, 2, 9, 6, 9, 7, the number 9 is the mode because it appears the most frequently. Measurement This section will review the basics of measurement systems used in the United States (sometimes called custom- ary measurement) and other countries, methods of performing mathematical operations with units of meas- urement, and the process of converting between different units. The use of measurement enables a connection to be made between mathematics and the real world. To meas- ure any object, assign a number and a unit of measure. For instance, when a fish is caught, it is often weighed in ounces and its length measured in inches. The following lesson will help you become more familiar with the types, conversions, and units of measurement. Types of Measurements The types of measurements used most frequently in the United States are listed below: Units of Length 12 inches (in) = 1 foot (ft) 3 feet = 36 inches = 1 yard (yd) 5,280 feet = 1,760 yards = 1 mile (mi) Units of Volume 8 ounces* (oz) = 1 cup (c) 2 cups = 16 ounces = 1 pint (pt) 2 pints = 4 cups = 32 ounces = 1 quart (qt) 4 quarts = 8 pints = 16 cups = 128 ounces = 1 gallon (gal) Units of Weight 16 ounces* (oz) = 1 pound (lb) 2,000 pounds = 1 ton (T) – THEA MATH REVIEW– 112 Units of Time 60 seconds (sec) = 1 minute (min) 60 minute = 1 hour (hr) 24 hours = 1 day 7 days = 1 week 52 weeks = 1 year (yr) 12 months = 1 year 365 days = 1 year *Notice that ounces are used to measure the dimensions of both volume and weight. Converting Units When performing mathematical operations, it may be necessary to convert units of measure to simplify a prob- lem. Units of measure are converted by using either multiplication or division. ■ To convert from a larger unit into a smaller unit, multiply the given number of larger units by the number of smaller units in only one of the larger units: (given number of the larger units) ϫ (the number of smaller units per larger unit) = answer in smaller units For example, to find the number of inches in 5 feet, multiply 5, the number of larger units, by 12, the number of inches in one foot: 5 feet = ? inches 5 feet ϫ 12 (the number of inches in a single foot) = 60 inches: 5 ft ϫ ᎏ 1 1 2 f i t n ᎏ = 60 in Therefore, there are 60 inches in 5 feet. Example Change 3.5 tons to pounds. 3.5 tons = ? pounds 3.5 tons ϫ ᎏ 2,00 1 0 t p o o n unds ᎏ = 7,000 pounds Therefore, there are 7,000 pounds in 3.5 tons. – THEA MATH REVIEW– 113 ■ To change a smaller unit to a larger unit, divide the given number of smaller units by the number of smaller units in only one of the larger units: = answer in larger units For example, to find the number of pints in 64 ounces, divide 64, the number of smaller units, by 16, the number of ounces in one pint. 64 ounces = ? pints = 4 pints Therefore, 64 ounces equals four pints. Example Change 32 ounces to pounds. 32 ounces = ? pounds = 2 pounds Therefore, 32 ounces equals two pounds. Basic Operations with Measurement You may need to add, subtract, multiply, and divide with measurement. The mathematical rules needed for each of these operations with measurement follow. ADDITION WITH MEASUREMENTS To add measurements, follow these two steps: 1. Add like units. 2. Simplify the answer by converting smaller units into larger units when possible. Example Add 4 pounds 5 ounces to 20 ounces. 4 lb 5 oz Be sure to add ounces to ounces. + 20 o z 4 lb 25 oz Because 25 ounces is more than 16 ounces (1 pound), simplify by dividing by 16: 1 lb r 9 oz 16 ozͤ25 ෆ o ෆ z ෆ –16 oz 9 oz 32 ounces ᎏᎏᎏ 16 ounces per pound 64 ounces ᎏᎏ 16 ounces per pint given number of smaller units ᎏᎏᎏᎏᎏ the number of smaller units per larger unit – THEA MATH REVIEW– 114 Then add the 1 pound to the 4 pounds: 4 pounds 25 ounces = 4 pounds + 1 pound 9 ounces = 5 pounds 9 ounces SUBTRACTION WITH MEASUREMENTS 1. Subtract like units if possible. 2. If not, regroup units to allow for subtraction. 3. Write the answer in simplest form. For example, 6 pounds 2 ounces subtracted from 9 pounds 10 ounces. 9 lb 10 oz Subtract ounces from ounces. – 6 lb 2 o z Then subtract pounds from pounds. 3 lb 8 oz Sometimes, it is necessary to regroup units when subtracting. Example Subtract 3 yards 2 feet from 5 yards 1 foot. Because 2 feet cannot be taken from 1 foot, regroup 1 yard from the 5 yards and convert the 1 yard to 3 feet. Add 3 feet to 1 foot. Then subtract feet from feet and yards from yards: 5 4 yd 1 4 ft – 3 y d 2 ft 1 yd 2ft 5 yards 1 foot – 3 yards 2 feet = 1 yard 2 feet MULTIPLICATION WITH MEASUREMENTS 1. Multiply like units if units are involved. 2. Simplify the answer. Example Multiply 5 feet 7 inches by 3. 5 ft 7 in Multiply 7 inches by 3, then multiply 5 feet by 3. Keep the units separate. ϫ 3 15 ft 21 in Since 12 inches = 1 foot, simplify 21 inches. 15 ft 21 in = 15 ft + 1 ft 9 in = 16 ft 9 in – THEA MATH REVIEW– 115 . number of inches in 5 feet, multiply 5, the number of larger units, by 12, the number of inches in one foot: 5 feet = ? inches 5 feet ϫ 12 (the number of inches in a single foot) = 60 inches: 5 ft. the quantity of numbers in the set. Average = (sum of set) ÷ (quantity of set) Example Find the average of 9, 4, 7, 6, and 4. ᎏ 9+4+7 5 +6+4 ᎏ = ᎏ 3 5 0 ᎏ = 6 The mean, or average, of the set. answer. Example Multiply 5 feet 7 inches by 3. 5 ft 7 in Multiply 7 inches by 3, then multiply 5 feet by 3. Keep the units separate. ϫ 3 15 ft 21 in Since 12 inches = 1 foot, simplify 21 inches. 15 ft 21 in = 15 ft