5.5 Averages and Probability

Learning Objectives

By the end of this section, you will be able to:

  • Calculate the mean of a set of numbers
  • Find the median of a set of numbers
  • Find the mode of a set of numbers
  • Apply the basic definition of probability

One application of decimals that arises often is finding the average of a set of numbers. What do you think of when you hear the word average? Is it your grade point average, the average rent for an apartment in your city, the batting average of a player on your favorite baseball team? The average is a typical value in a set of numerical data. Calculating an average sometimes involves working with decimal numbers. In this section, we will look at three different ways to calculate an average.

Calculate the Mean of a Set of Numbers

The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.

Suppose Ethan’s first three test scores were 85,88,and94.To find the mean score, he would add them and divide by 3.

His mean test score is 8989 points.

Did you notice that in the last example, while all the numbers were whole numbers, the mean was 59.5,59.5, a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money, and it will make sense to report the mean in dollars and cents.

Identify the Mode of a Set of Numbers

The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average. Yes, it can be confusing when the word average refers to two different numbers, the mean and the median! In fact, there is a third number that is also an average. This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.

Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.

Use the Basic Definition of Probability

The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals.

For example, picture a fruit bowl that contains five pieces of fruit – three bananas and two apples.

If you want to choose one piece of fruit to eat for a snack and don’t care what it is, there is a 3535 probability you will choose a banana, because there are three bananas out of the total of five pieces of fruit. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.

Two equations are shown. The top equation says the probability of an event equals the number of favorable outcomes over the total number of outcomes. The bottom equation says the probability of choosing a banana equals 3 over 5. There is a blue arrow pointing to the 3 with the text, 'There are 3 bananas.' There is a blue arrow pointing to the 5 with the text, 'There are 5 pieces of fruit.'

Converting the fraction 35 to a decimal, we would say there is a 0.6 probability of choosing a banana.

This basic definition of probability assumes that all the outcomes are equally likely to occur. If you study probabilities in a later math class, you’ll learn about several other ways to calculate probabilities.

Section 5.5 Exercises

Practice Makes Perfect

Calculate the Mean of a Set of Numbers

In the following exercises, find the mean.

  1. 3 , 8 , 2 , 2 , 5
  2. 6 , 1 , 9 , 3 , 4 , 7
  3. 65 , 13 , 48 , 32 , 19 , 33
  4. 34 , 45 , 29 , 61 , and 41
  5. 202 , 241 , 265 , 274
  6. 525 , 532 , 558 , 574
  7. 12.45 , 12.99 , 10.50 , 11.25 , 9.99 , 12.72
  8. 28.8 , 32.9 , 32.5 , 27.9 , 30.4 , 32.5 , 31.6 , 32.7
  9. Four girls leaving a mall were asked how much money they had just spent. The amounts were $0 , $14.95 , $35.25 , and $25.16 . Find the mean amount of money spent.
  10. Juan bought 5 shirts to wear to his new job. The costs of the shirts were $32.95 , $38.50 , $30.00 , $17.45 , and $24.25 . Find the mean cost.
  11. The number of minutes it took Jim to ride his bike to school for each of the past six days was 21 , 18 , 16 , 19 , 24 , and 19 . Find the mean number of minutes.
  12. Norris bought six books for his classes this semester. The costs of the books were $74.28 , $120.95 , $52.40 , $10.59 , $35.89 , and $59.24 . Find the mean cost.
  13. The top eight hitters in a softball league have batting averages of .373 , .360 , .321 , .321 , .320 , .312 , .311 , and .311 . Find the mean of the batting averages. Round your answer to the nearest thousandth.
  14. The monthly snowfall at a ski resort over a six-month period was 60.3, 79.7, 50.9, 28.0, 47.4, and 46.1 inches. Find the mean snowfall.
    Find the Median of a Set of Numbers

In the following exercises, find the median.

  1. 24 , 19 , 18 , 29 , 21
  2. 48 , 51 , 46 , 42 , 50
  3. 65 , 56 , 35 , 34 , 44 , 39 , 55 , 52 , 45
  4. 121 , 115 , 135 , 109 , 136 , 147 , 127 , 119 , 110
  5. 4 , 8 , 1 , 5 , 14 , 3 , 1 , 12
  6. 3 , 9 , 2 , 6 , 20 , 3 , 3 , 10
  7. 99.2 , 101.9 , 98.6 , 99.5 , 100.8 , 99.8
  8. 28.8 , 32.9 , 32.5 , 27.9 , 30.4 , 32.5 , 31.6 , 32.7
  9. Last week Ray recorded how much he spent for lunch each workday. He spent $6.50 , $7.25 , $4.90 , $5.30 , and $12.00 . Find the median.
  10. Michaela is in charge of 6 two-year olds at a daycare center. Their ages, in months, are 25 , 24 , 28 , 32 , 29 , and 31 . Find the median age.
  11. Brian is teaching a swim class for 6 three-year olds. Their ages, in months, are 38,41,45,36,40,and42. Find the median age.
  12. Sal recorded the amount he spent for gas each week for the past 8 weeks. The amounts were $38.65, $32.18, $40.23, $51.50, $43.68, $30.96, $41.37, and $44.72. Find the median amount.
    Identify the Mode of a Set of Numbers

In the following exercises, identify the mode.

  1. 2 , 5 , 1 , 5 , 2 , 1 , 2 , 3 , 2 , 3 , 1
  2. 8 , 5 , 1 , 3 , 7 , 1 , 1 , 7 , 1 , 8 , 7
  3. 18 , 22 , 17 , 20 , 19 , 20 , 22 , 19 , 29 , 18 , 23 , 25 , 22 , 24 , 23 , 22 , 18 , 20 , 22 , 20
  4. 42 , 28 , 32 , 35 , 24 , 32 , 48 , 32 , 32 , 24 , 35 , 28 , 30 , 35 , 45 , 32 , 28 , 32 , 42 , 42 , 30
  5. The number of children per house on one block: 1 , 4 , 2 , 3 , 3 , 2 , 6 , 2 , 4 , 2 , 0 , 3 , 0.
  6. The number of movies watched each month last year: 2 , 0 , 3 , 0 , 0 , 8 , 6 , 5 , 0 , 1 , 2 , 3.
  7. The number of units being taken by students in one class: 12 , 5 , 11 , 10 , 10 , 11 , 5 , 11 , 11 , 11 , 10 , 12 .
  8. The number of hours of sleep per night for the past two weeks: 8 , 5 , 7 , 8 , 8 , 6 , 6 , 6 , 6 , 9 , 7 , 8 , 8 , 8 .
    Use the Basic Definition of Probability

In the following exercises, express the probability as both a fraction and a decimal. (Round to three decimal places, if necessary.)

  1. Josue is in a book club with 20 members. One member is chosen at random each month to select the next month’s book. Find the probability that Josue will be chosen next month.
  2. Jessica is one of eight kindergarten teachers at Mandela Elementary School. One of the kindergarten teachers will be selected at random to attend a summer workshop. Find the probability that Jessica will be selected.
  3. There are 24 people who work in Dane’s department. Next week, one person will be selected at random to bring in doughnuts. Find the probability that Dane will be selected. Round your answer to the nearest thousandth.
  4. Monica has two strawberry yogurts and six banana yogurts in her refrigerator. She will choose one yogurt at random to take to work. Find the probability Monica will choose a strawberry yogurt.
  5. Michel has four rock CDs and six country CDs in his car. He will pick one CD to play on his way to work. Find the probability Michel will pick a rock CD.
  6. Noah is planning his summer camping trip. He can’t decide among six campgrounds at the beach and twelve campgrounds in the mountains, so he will choose one campground at random. Find the probability that Noah will choose a campground at the beach.
  7. Donovan is considering transferring to a 4-year college. He is considering 10 out-of state colleges and 4 colleges in his state. He will choose one college at random to visit during spring break. Find the probability that Donovan will choose an out-of-state college.
  8. There are 258,890,850 number combinations possible in the Mega Millions lottery. One winning jackpot ticket will be chosen at random. Brent chooses his favorite number combination and buys one ticket. Find the probability Brent will win the jackpot. Round the decimal to the first digit that is not zero, then write the name of the decimal.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
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ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?