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Lesson 48 ! page 1

Lesson 48*

Optional Percent Topics The three topics included here are ! Probability ! Pie Charts ! Markup and Margin Instead of homework, there are three worksheets, one for each topic. There is no homework or lab for this section.

Probability

Probabilities are often reported using percents. Calculating Probability You can’t predict the future, but you can calculate the probability of some events. The study of probability began with a gambler asking a mathematician for some advice, and games of chance are an easy way to begin studying what is meant by probability.

Suppose instead of an ordinary die, the six faces of your special die are the ones shown unfolded below right. It’s possible to roll any number 1 through 5, but there is no 6. Instead there are two faces with the number 3. Does this die have five, or six possible outcomes? Since it has six faces, each equally likely, there are still six possible outcomes. We distinguish between the two ways of rolling a three, treating them as two different outcomes.

© 2010 Cheryl Wilcox

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Lesson 48 ! page 2

Finding all the Outcomes A couple has three children. What is the probability that they are all boys? We assume that girls and boys are equally likely. This problem is similar to asking, “If you toss three coins, what is the probability that all three coins land tails?” The tricky part here is to figure out how many possible outcomes there are. Let’s list all the possible tosses of three coins. Note that we keep track of each coin separately, as if we tossed a penny, a dime, and a nickel. The first row is the possibilities when the first coin is heads; the second row is the possibilities with the first coin tails: HHH THH

HHT THT

HTH TTH

HTT TTT

TTH There are eight possible outcomes, only one of which has three tails. The probability of 3 tails is 1/8 = 0.125 = 12.5%. Example: Three coins are tossed. What is the probability that at least two are heads? Look at the possible outcomes and find all those with at least two heads: HHH THH

HHT THT

HTH TTH

HTT TTT

Four of the eight possible outcomes have at least two heads. The probability is 4/8 = 0.5 = 50% Other Ways of Estimating a Probability This simple and exact way of calculating probability shown above can’t always be applied. For example, when a batter walks up to the plate, what is the probability that he or she will get a hit? There are two possible outcomes: OUT, or HIT, but the outcomes are not equally likely, so we cannot calculate as above. Instead, we assume that the future will be similar to the past. The player’s batting average tells the ratio of hits to at bats for the past, and we suppose that the batter will play at about the same skill level in the future. So if a player’s batting average is 0.3125, we may say that the probability of getting a hit is 31.25%. But this is a prediction based on past performance, not a probability based on physics like that in rolling a die. Yet another kind of probability is based on a person’s judgment. If the player tells you ahead of time “I’m 90% sure I’ll get a hit,” this is called a confidence probability, estimated based on the person’s experience and expertise, not by using a formula.

© 2010 Cheryl Wilcox

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Lesson 48 ! page 3

Pie Charts

Graphic representations give people a “feeling” for information in a way that numbers often don’t. Percents of the Same Base A pie chart shows how some whole is divided. The whole circle of the “pie” is divided into parts that show the proportions of their relationship. The parts of the pie could be thought of as fractions that add up to one whole, or as percents that add up to 100%.

Pie Chart Jokes This one is perhaps more revealing of the author than reliable information about general use.

Example: Fill in the missing percent in the pie chart. The percents must add up to 100%. 50 + 30 + 10 + 8 + x = 100 98 + x = 100 x=2 The missing percent is 2%. Making a Pie Chart If you’re using a computer, you can make a pie chart in most spreadsheet programs. There are also websites that provide software for making pie charts, such as http://www.shodor.org/interactivate/activities/PieChart/. It’s instructive to make a few by hand, but it gets tedious, so it’s nice to know how to make the charts with software. Example: Make a pie chart to represent the data about the ages of DVC students in Fall 2008.

The circle is divided into 8 equal parts. Each part is 1/8 of the whole circle. 1/8 = 0.125 = 12.5% We estimate the percents in the chart relative to the 12.5%, 25%, 37.5%, etc. Choose a different color for each section of the chart.

Age 19 or less: 30.8% more than 25%, less than 37.5% Age 20 – 24: 31.4% similar Age 25 – 34: 17% more than 12.5%, less than 25%

Age 35 – 39: 12.1% very slightly less than 12.5% Age 50+: 8.1% The estimates show the relative proportions, even though the measurements are not exact.

Notice that the pie chart does not tell us the size of the whole. It only shows the sizes of the various age groups relative to one another and to the whole. It’s good practice to identify the base of any percent you are using so that the reader can calculate the numbers in the chart.

© 2010 Cheryl Wilcox

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Lesson 48 ! page 4

Markup and Margin

Retail sales uses two calculations with percents to talk about the difference between the wholesale price and the sales price of goods and services. “Markup vs Margin: Is there a difference? Absolutely... Terminology speaking, markup percentage is the percentage difference between the actual cost and the selling price, while gross margin percentage is the percentage difference between the selling price and the profit.” WikiCFO “While markup and margin refer to the same thing, markup is calculated based on the relationship of gross profit to cost while margin is calculated based on the relationship of gross profit to revenue.” Microsoft Office Support “The golden rule to start you off is this: markup belongs to cost and margin belongs to sales.” FortuneCity Just to investigate what they’re talking about above, let’s imagine we have an item that cost $40 wholesale and is being sold for $100. cost + markup = price

The difference between the cost and selling price is the (gross) profit, also called the markup, here $60. The markup percentage is the markup as a percent of the cost. markup / cost = 60 / 40 = 1.5 = 150% markup The margin is the markup as a percent of the sales price. profit / sale price = 60 / 100 = 0.6 = 60% margin The numerator of both fractions is the profit = markup. Only the denominator, the base of the percent, is different in each case. That’s why “markup belongs to cost and margin belongs to sales.” The denominator for the markup percent is the cost and the denominator for the margin is the sale price. Since the sale price is higher than the wholesale price, the margin is always a smaller percent than the markup. Example: Find the missing information. 1. An item cost $20 wholesale and will be sold for $40. What is the markup, the markup percentage and the margin? markup is $20 markup percentage is $20/$20 = 1 = 100% margin is $20/$40 = 0.5 = 50%

© 2010 Cheryl Wilcox

2. An item cost $50 wholesale and the markup percentage is 150%. What is the sale price and the margin?

3. An item with a 50% markup percentage is being sold for $100. What is the cost, markup, and margin?

150% of $50 is the markup. 1.5 • $50 = $75

The sale price is 150% of the cost 1.5 • cost = 100 cost = $100/1.5 = $66.67

The sale price is the cost + the markup $50 + $75 = $125. The margin is 75/125 = 0.6 60% margin

markup is $100 – $66.67 = $33.33 margin is $33.33/$100 = 33.33%

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Lesson 48 ! page 5

The margin and markup are closely related. You might have noticed that a markup of 150% resulted in a margin of 60% in two different examples. This is no accident. The margin can be calculated directly from the markup percentage and vice versa, and you can find tables relating the two percents on business websites. We can figure it out ourselves. Start with cost + markup = price Both markup and price can be written as percents of cost: markup = (markup percent) • cost price = (100% + markup percent) • cost Even if you don’t know the cost, the margin is the ratio of markup to price, so the cost cancels in the fraction.

margin =

markup price

=

(markup percent) • cost (100% + markup percent) • cost

This gives us the formula

margin =

markup percent 100% + markup percent

Example: Find the missing information. A business uses a markup of 30%. What is the margin?

A business uses a markup of 200%. What is the margin?

30% / 130% = 0.23076…

200% / 300% = 0.6666…

margin is about 23%

about 67%

The markup can be (and often is) more than 100%, but the margin is below 100% if the item cost anything at all. The margin measures the profit as a part of the sale, and since it is only a part, it is less than 100%.

!

© 2010 Cheryl Wilcox

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Lesson 48 ! page 6

Lesson 48: Optional Percent Topics

Worksheet 1: Probability

Name__________________________________ The experiment is to roll two dice and add their face values. Fill in the chart with all the possible outcomes by adding the values of the dice across the row and column.

Calculate the probability of rolling the sums listed below. Probability = number of ways to get the sum / total number of ways dice can land

1 2 3 4 5 6 7 8 9 10 11 12 a sum less than 6 a sum greater than 6

© 2010 Cheryl Wilcox

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Lesson 48 ! page 6a

Lesson 48: Optional Percent Topics

Worksheet 1: Probability Answers The experiment is to roll two dice and add their face values. Fill in the chart with all the possible outcomes by adding the values of the dice across the row and column.

2

3

4

5

6

7

3

4

5

6

7

8

4

5

6

7

8

9

5

6

7

8

9

10

6

7

8

9

10

11

7

8

9

10

11

12

Calculate the probability of rolling the sums listed below. Probability = number of ways to get the sum / total number of ways dice can land

1

0/36 = 0

impossible

2

1/36 = 0.27777…

3

2/36 = 0.055555…. about 5.6%

4

3/36 = 0.083333…. about 8.3%

about 2.8%

5

4/36 = 0.11111…

6

5/36 = 0.138888…. about 13.9%

7

6/36 = 0.16666….

8

5/36 = 0.138888…. about 13.9%

9

4/36 = 0.11111…

about 11.1%

about 16.7%

about 11.1%

10

3/36 = 0.083333…. about 8.3%

11

2/36 = 0.055555…. about 5.6%

12 a sum less than 6 a sum greater than 6

© 2010 Cheryl Wilcox

1/36 = 0.27777…

about 2.8%

10/36 = 0.27777…. about 27.8% 21/36 = 0.58333333…. about 58.3%

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Lesson 48: Optional Percent Topics

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