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Fractions, decimals and percents

Probability · level 4, lesson 3 of 6

A question to start

The weather forecast says: "40% chance of rain tomorrow." What does that actually mean?

  • It will rain for 40% of the day?
  • It will rain on 40% of the town?
  • 40% of forecasters think it will rain?
  • Something else?

Ask a few adults and you will get different answers. Then a second question: tomorrow it does not rain. Was the forecast wrong?

By the end of this lesson you will be able to answer both, and to move easily between the three ways probabilities are written: as fractions, as decimals and as percents.

The idea

One number, three costumes

A probability is a number from 0 to 1. The same number can be written in three ways, like one person wearing three different costumes:

  • a fraction: \(\tfrac25\)
  • a decimal: \(0.4\)
  • a percent: \(40\%\)

All three say exactly the same thing: 2 parts out of every 5, or 4 out of every 10, or 40 out of every 100.

0 0 0% 1/4 0.25 25% 2/5 0.4 40% 1/2 0.5 50% 3/4 0.75 75% 1 1 100%

Fractions are best when you are counting outcomes (\(\tfrac{3}{8}\) from a tree). Decimals are best for comparing and for calculators. Percents are what you see in the news, in weather forecasts and on game shows, because "out of 100" is easy to picture.

Changing between them

Fraction to decimal: divide. The fraction bar means "divided by". \(\tfrac25 = 2 \div 5 = 0.4\), and \(\tfrac38 = 3 \div 8 = 0.375\). Sometimes it is quicker to make the bottom 10 or 100 first: \(\tfrac{2}{5} = \tfrac{4}{10} = 0.4\); \(\tfrac{7}{20} = \tfrac{35}{100} = 0.35\).

Decimal to percent: multiply by 100. "Percent" means "per hundred" (from the Latin per centum). So 0.4 is 40 per hundred, \(40\%\). And \(0.375 = 37.5\%\).

Percent to fraction: put it over 100 and simplify. \(40\% = \tfrac{40}{100} = \tfrac25\). \(5\% = \tfrac{5}{100} = \tfrac{1}{20}\).

Some fractions never stop as decimals. \(\tfrac13 = 0.333\ldots\) with 3s forever, so we round: about 0.33, or about \(33.3\%\). That is fine for comparing, but keep the fraction if you need the exact answer.

The scale from 0 to 1 becomes the scale from \(0\%\) to \(100\%\). Impossible is \(0\%\); certain is \(100\%\); even chance is \(50\%\). A chance of "\(110\%\)" is impossible, however much a sports coach asks for it.

Comparing chances

Which is most likely: \(\tfrac38\), \(0.4\) or \(35\%\)? In different costumes it is hard to tell. Put them all in the same costume, decimals:

\[\tfrac38 = 0.375, \qquad 0.4, \qquad 35\% = 0.35\]

Now order them: \(0.35 < 0.375 < 0.4\). So \(0.4\) is the most likely and \(35\%\) the least.

"1 in 20"

News stories often say "a 1 in 20 chance". That is the fraction \(\tfrac{1}{20}\), which is \(0.05\) or \(5\%\). Going the other way, a \(25\%\) chance is "1 in 4". This form is handy for small chances: "1 in 1000" is easier to picture than \(0.1\%\).

What "40% chance of rain" means

Here is a picture to hold in your head. Imagine every day, over many years, on which the forecast for your town was "40% chance of rain". Put one square for each of 100 such days.

100 days with the forecast “40% chance of rain”: about 40 of them get rain, about 60 stay dry

A 40% chance of rain means: on days like this, it rains (at least a little, at your place) on about 40 out of every 100 of them. It does not say how long it will rain, or how hard. It says how often days like tomorrow turn out wet.

So which answers at the start were right? Not "40% of the day", and not "40% of forecasters". It is a probability, a long-run fraction, just like the coin flips in level 3.

And was the forecast wrong if it stays dry? No. A 40% forecast expects most such days to be dry: 60 out of 100. One dry day cannot show a forecast was wrong, just as one tails cannot show a coin is unfair.

How then do you check a forecaster? Look at many days. Gather all the days on which she said 40%, and count how many were rainy. If it is close to 40% of them, her 40% means what it says. Do the same for her 10% days, her 70% days, and so on. A forecaster whose numbers match what happens over many days is called well calibrated. That is the only fair way to judge a probability: by what happens in the long run.

The opposite, in percents

The rule \(P(\text{not } A) = 1 - P(A)\) from level 3 becomes "100% minus" in percents. A 40% chance of rain is a \(100\% - 40\% = 60\%\) chance of no rain.

Worked examples

1. One probability, three ways. In three coin flips, \(P(\text{exactly two heads}) = \tfrac38\) (from lesson 2).

  • Decimal: \(3 \div 8 = 0.375\).
  • Percent: \(0.375 \times 100 = 37.5\%\).

2. Ordering chances. Put in order, least likely first: \(\tfrac{7}{20}\), \(0.33\), \(\tfrac13\), \(36\%\).

As decimals: \(\tfrac{7}{20} = 0.35\); \(0.33\); \(\tfrac13 = 0.333\ldots\); \(36\% = 0.36\).

Order: \(0.33 < 0.333\ldots < 0.35 < 0.36\), so \(0.33\), then \(\tfrac13\), then \(\tfrac{7}{20}\), then \(36\%\). Notice that \(0.33\) and \(\tfrac13\) are not equal: \(\tfrac13\) is a tiny bit more.

3. Two forecasts. Monday's forecast is a 40% chance of rain; Tuesday's is 70%.

  • Tuesday is more likely to be rainy.
  • \(P(\text{Tuesday dry}) = 100\% - 70\% = 30\%\).
  • "70%" can also be said "7 in 10", or \(\tfrac{7}{10}\).

4. Dice in percents. Two dice: \(P(\text{total } 7) = \tfrac{6}{36} = \tfrac16\). As a decimal \(1 \div 6 = 0.1666\ldots\), about \(0.167\), which is about \(16.7\%\). \(P(\text{double six}) = \tfrac{1}{36}\), about \(0.028\), which is about \(2.8\%\): "1 in 36".

5. Checking a forecaster. On 30 days a forecaster said "40% chance of rain". It rained on 11 of them.

\(\tfrac{11}{30} = 0.3666\ldots\), about \(36.7\%\). That is close to \(40\%\) for only 30 days. If it had rained on 25 of the 30 days (about \(83\%\)), her 40% would look far too low.

Try it

  1. Write each as a decimal and as a percent: (a) \(\tfrac14\); (b) \(\tfrac15\); (c) \(\tfrac{3}{10}\); (d) \(\tfrac78\); (e) \(\tfrac13\) (round to one decimal place in the percent).
  2. Write each percent as a fraction in simplest form: (a) \(60\%\); (b) \(5\%\); (c) \(12.5\%\).
  3. Put in order from least to most likely: \(0.6\), \(\tfrac58\), \(58\%\), \(\tfrac35\).
  4. The forecast is a 25% chance of snow. What is the chance of no snow, as a percent and as a fraction?
  5. A game show host says your chance of winning is "1 in 8". Your friend's chance in another game is 12%. Who has the better chance?
  6. A coin is flipped. Write \(P(\text{heads})\) in all three forms. Then write \(P(\text{three heads in three flips})\) in all three forms.
  7. On 50 days a forecaster said "20% chance of rain", and it rained on 9 of them. Is her 20% about right?
  8. Ravi says: "The forecast said 90% chance of rain and it stayed dry, so the forecast was wrong." Is he right?
Answers
  1. (a) \(0.25 = 25\%\). (b) \(0.2 = 20\%\). (c) \(0.3 = 30\%\). (d) \(7 \div 8 = 0.875 = 87.5\%\). (e) \(0.333\ldots\), about \(33.3\%\).
  2. (a) \(\tfrac{60}{100} = \tfrac35\). (b) \(\tfrac{5}{100} = \tfrac{1}{20}\). (c) \(\tfrac{12.5}{100} = \tfrac{125}{1000} = \tfrac18\).
  3. As decimals: \(0.6\), \(0.625\), \(0.58\), \(0.6\). So \(58\%\), then \(0.6\) and \(\tfrac35\) (equal), then \(\tfrac58\).
  4. \(100\% - 25\% = 75\%\), which is \(\tfrac34\).
  5. "1 in 8" is \(\tfrac18 = 0.125 = 12.5\%\), which is more than \(12\%\). You have the (slightly) better chance.
  6. \(P(\text{heads}) = \tfrac12 = 0.5 = 50\%\). Three heads: \(\tfrac18 = 0.125 = 12.5\%\).
  7. \(\tfrac{9}{50} = 0.18 = 18\%\). That is close to \(20\%\); her forecasts look about right.
  8. Not on this evidence. A 90% chance means that on about 10 in every 100 such days it stays dry. One dry day is allowed. If it stayed dry on many of her 90% days (say 40 out of 100), then her 90% would be wrong.

Common mistake

Thinking a forecast was "wrong" because the less likely thing happened. A 70% chance of rain is a 30% chance of dry. Dry days will happen. Probabilities are judged over many days, not one.

Thinking 0.33 equals \(\tfrac13\). Rounded decimals are close, not exact. When comparing close chances, use the fractions, or more decimal places.

Where it leads