5.9: Averages and Probability (Part 2)
- Page ID
- 6453
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\dfrac{3}{5}\) 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.
The probability of an event is the number of favorable outcomes divided by the total number of outcomes possible.
\[Probability = \dfrac{number\; of\; favorable\; outcomes}{total\; number\; of\; outcomes}\]
Converting the fraction \(\dfrac{3}{5}\) to a decimal, we would say there is a 0.6 probability of choosing a banana.
Probability of choosing a banana = \(\dfrac{3}{5}\)
Probability of choosing a banana = 0.6
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.
The ski club is holding a raffle to raise money. They sold 100 tickets. All of the tickets are placed in a jar. One ticket will be pulled out of the jar at random, and the winner will receive a prize. Cherie bought one raffle ticket. (a) Find the probability she will win the prize. (b) Convert the fraction to a decimal.
Solution
(a)
| What are you asked to find? | The probability Cherie wins the prize. |
| What is the number of favorable outcomes? | 1, because Cherie has 1 ticket. |
| Use the definition of probability. | $$Probability\; of\; an\; event = \dfrac{number\; of\; favorable\; outcomes}{total\; number\; of\; outcomes}$$ |
| Substitute into the numerator and denominator. | Probability Cherie wins = \(\dfrac{1}{100}\) |
(b)
| Write the probability as a fraction. | Probability = \(\dfrac{1}{100}\) |
| Convert the fraction to a decimal. | Probability = 0.01 |
Ignaly is attending a fashion show where the guests are seated at tables of ten. One guest from each table will be selected at random to receive a door prize. (a) Find the probability Ignaly will win the door prize for her table. (b) Convert the fraction to a decimal.
- Answer a
-
\(\frac{1}{10}\)
- Answer b
-
\(0.1\)
Hoang is among 20 people available to sit on a jury. One person will be chosen at random from the 20. (a) Find the probability Hoang will be chosen. (b) Convert the fraction to a decimal.
- Answer a
-
\(\frac{1}{20}\)
- Answer b
-
\(0.05\)
Three women and five men interviewed for a job. One of the candidates will be offered the job. (a) Find the probability the job is offered to a woman. (b) Convert the fraction to a decimal.
Solution
| What are you asked to find? | The probability the job is offered to a woman. |
| What is the number of favorable outcomes? | 3, because there are three women. |
| What are the total number of outcomes? | 8, because 8 people interviewed. |
| Use the definition of probability. | $$Probability\; of\; an\; event = \dfrac{number\; of\; favorable\; outcomes}{total\; number\; of\; outcomes}$$ |
| Substitute into the numerator and denominator. | Probability = \(\dfrac{3}{8}\) |
(b)
| Write the probability as a fraction. | Probability = \(\dfrac{3}{8}\) |
| Convert the fraction to a decimal. | Probability = 0.375 |
A bowl of Halloween candy contains 5 chocolate candies and 3 lemon candies. Tanya will choose one piece of candy at random.(a) Find the probability Tanya will choose a chocolate candy.(b) Convert the fraction to a decimal.
- Answer a
-
\(\frac{5}{8}\)
- Answer b
-
\(0.625\)
Dan has 2 pairs of black socks and 6 pairs of blue socks. He will choose one pair at random to wear tomorrow. (a) Find the probability Dan will choose a pair of black socks (b) Convert the fraction to a decimal.
- Answer a
-
\(\frac{2}{8}\)
- Answer b
-
\(0.25\)
Mean, Median, and Mode
Find the Mean of a Data Set
Find the Median of a Data Set
Find the Mode of a Data Set
Practice Makes Perfect
Calculate the Mean of a Set of Numbers
In the following exercises, find the mean.
- 3, 8, 2, 2, 5
- 6, 1, 9, 3, 4, 7
- 65, 13, 48, 32, 19, 33
- 34, 45, 29, 61, and 41
- 202, 241, 265, 274
- 525, 532, 558, 574
- 12.45, 12.99, 10.50, 11.25, 9.99, 12.72
- 28.8, 32.9, 32.5, 27.9, 30.4, 32.5, 31.6, 32.7
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 24, 19, 18, 29, 21
- 48, 51, 46, 42, 50
- 65, 56, 35, 34, 44, 39, 55, 52, 45
- 121, 115, 135, 109, 136, 147, 127, 119, 110
- 4, 8, 1, 5, 14, 3, 1, 12
- 3, 9, 2, 6, 20, 3, 3, 10
- 99.2, 101.9, 98.6, 99.5, 100.8, 99.8
- 28.8, 32.9, 32.5, 27.9, 30.4, 32.5, 31.6, 32.7
- 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.
- 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.
- Brian is teaching a swim class for 6 three-year olds. Their ages, in months, are 38, 41, 45, 36, 40, and 42. Find the median age.
- 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.
- 2, 5, 1, 5, 2, 1, 2, 3, 2, 3, 1
- 8, 5, 1, 3, 7, 1, 1, 7, 1, 8, 7
- 18, 22, 17, 20, 19, 20, 22, 19, 29, 18, 23, 25, 22, 24, 23, 22, 18, 20, 22, 20
- 42, 28, 32, 35, 24, 32, 48, 32, 32, 24, 35, 28, 30, 35, 45, 32, 28, 32, 42, 42, 30
- The number of children per house on one block: 1, 4, 2, 3, 3, 2, 6, 2, 4, 2, 0, 3, 0.
- The number of movies watched each month last year: 2, 0, 3, 0, 0, 8, 6, 5, 0, 1, 2, 3.
- The number of units being taken by students in one class: 12, 5, 11, 10, 10, 11, 5, 11, 11, 11, 10, 12.
- 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.)
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
Everyday Math
- Joaquin gets paid every Friday. His paychecks for the past 8 Fridays were $315, $236.25, $236.25, $236.25 $315, $315, $236.25, $393.75. Find the (a) mean (b) median, and (c) mode.
- The cash register receipts each day last week at a coffee shop were $1,845, $1,520, $1,438, $1,682, $1,850, $2,721, $2,539. Find the (a) mean, (b) median, and (c) mode.
Writing Exercises
- Explain in your own words the difference between the mean, median, and mode of a set of numbers.
- Make an example of probability that relates to your life. Write your answer as a fraction and explain what the numerator and denominator represent.
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) After looking at the checklist, do you think you are well prepared for the next section? Why or why not?
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."