One Probability, Three Common Forms
A probability can be written as a fraction, a decimal, or a percent. These are different ways to express the same number, not different probabilities. The form to use depends on the question or on what is easiest to interpret in context.
As discussed in Probability Is a Number Between 0 and 1, a probability is between 0 and 1, inclusive. A fraction such as \(9/40\), a decimal such as \(0.225\), and a percent such as \(22.5\%\) can all describe the same probability. The symbol \(\%\) means “per 100,” so \(22.5\%\) means \(22.5\) out of 100, or \(0.225\).
These rules work in either direction. For example, \(0.08\) as a percent is \(0.08\times100=8\%\). Conversely, \(8\%\) as a decimal is \(8\div100=0.08\). Likewise, \(3\%=0.03\), and \(0.07=7\%\). Keeping track of the direction of the conversion is essential: decimal-to-percent moves the decimal point two places to the right, while percent-to-decimal moves it two places to the left.
Convert First, Then Decide Whether to Round
Some conversions produce a terminating decimal, such as \(9/40=0.225\). Others produce a decimal that continues indefinitely. For instance, \(7/24=0.291666\ldots\), where the 6s continue. The fraction \(7/24\) is exact; the decimal written with a limited number of digits is an approximation.
Rounding replaces a value with a nearby value at a specified precision. “Round to three decimal places” means keep three digits after the decimal point and use the next digit to decide whether to increase the last kept digit. If the next digit is 5 or greater, round up; if it is less than 5, leave the last kept digit unchanged. For a percent, pay attention to whether the requested precision is decimal places in the percent, such as the nearest tenth of a percent, or decimal places in the probability itself.
Rounding too early can change the final answer. For example, if a decimal is rounded before it is converted to a percent, the rounded percent may differ slightly from the percent obtained using the original value. Retaining extra digits during the work avoids that problem. The requested form and precision should determine the final presentation, not the number of digits that happen to appear on a calculator screen.
Worked Example: A Fraction, Decimal, and Percent
Worked Example: A Fraction, Decimal, and Percent
A container holds 40 equally likely tokens, of which 9 are blue. One token is selected at random. Let \(B\) be the event that the selected token is blue. Express \(P(B)\) as a fraction, a decimal, and a percent.
Fraction: There are 9 favorable tokens out of 40 equally likely tokens, so \(P(B)=9/40\). This fraction is already in simplest form: 9 and 40 have no common factor greater than 1.
Decimal: Divide the numerator by the denominator.
Percent: Multiply the decimal by 100 and include the percent symbol.
Check: Convert \(22.5\%\) back to a decimal by dividing by 100: \(22.5\div100=0.225\). The fraction also gives \(9/40=0.225\), so all three forms agree.
Conclude: The probability of selecting a blue token is \(9/40\), or \(0.225\), or \(22.5\%\). These forms express the same chance.
Reporting a Repeating Decimal at the Requested Precision
When a decimal does not terminate, do not treat a short calculator display as the exact value. Keep the fraction as an exact form when useful, and use enough decimal digits to round accurately. Then convert to a percent if that is the form requested.
Worked Example: Rounding a Probability from a Fraction
A prize is assigned to one of 24 equally likely numbered spaces. Seven spaces award a small prize. Let \(S\) be the event that the chosen space awards the prize. Give \(P(S)\) as a decimal rounded to three decimal places and as a percent rounded to the nearest tenth of a percent.
Exact fraction: Seven of the 24 spaces award the prize, so \(P(S)=7/24\). Divide to get \(7\div24=0.291666\ldots\). For three decimal places, keep 0.291 and inspect the next digit, which is 6. Since 6 is at least 5, increase the thousandths digit from 1 to 2.
Percent: Use the decimal value before rounding: \(0.291666\ldots\times100=29.1666\ldots\%\). To round to the nearest tenth of a percent, keep one digit after the decimal point in the percent and inspect the next digit. That next digit is 6, so \(29.1\%\) rounds to \(29.2\%\).
Check: Converting \(29.2\%\) back to a decimal gives \(29.2\div100=0.292\). This agrees with the probability rounded to three decimal places.
Conclude: The exact probability is \(7/24\). To the requested precision, it is approximately \(0.292\), or \(29.2\%\). The approximation symbols matter because the decimal and percent have been rounded.
Notice that the exact fraction does not change when the decimal is rounded. It remains \(7/24\); \(0.292\) is a convenient approximation, not a new exact value. When the question requests an exact answer, a fraction such as \(7/24\) is often preferable to a rounded decimal.
Let the Context Set the Final Form
A context may ask for a probability as a decimal, a percent, or a fraction. Follow that request, and match its precision. A report describing a chance “out of 100” may be clearest as a percent. A calculation involving counts may naturally begin as a fraction. A probability model or calculator output may use decimals.
A rounded result should still make sense in context. A probability of \(0.03648\) is the same as \(3.648\%\). If a question asks for the percent to the nearest hundredth of a percent, the answer is \(3.65\%\). If it asks for the decimal to four decimal places, the answer is \(0.0365\). The two rounded answers are consistent because \(3.65\%\div100=0.0365\).
Worked Example: Choose the Requested Percent Precision
A fictional quality-control model assigns probability \(0.03648\) to a package requiring an additional inspection. Write this probability as a percent rounded to the nearest hundredth of a percent, and as a decimal rounded to four decimal places.
Percent: Multiply the original decimal by 100. The result is \(3.648\%\). The nearest hundredth of a percent means two digits after the percent’s decimal point. The third digit after the decimal point is 8, so round the hundredths digit up.
Decimal: For four decimal places, keep \(0.0364\) and inspect the next digit, which is 8. Increase the fourth decimal digit from 4 to 5.
Check and conclude: The rounded forms agree because \(3.65\%\div100=0.0365\). Thus the probability is \(3.65\%\), or \(0.0365\), at the requested precision. Both are approximations to the original probability \(0.03648\).
This example illustrates why “nearest hundredth of a percent” is not the same instruction as “nearest hundredth as a decimal.” The former asks for two digits after the decimal point in a number that is already a percent. Always identify which quantity is being rounded before choosing the digits to keep.
Common Mistakes and AP Exam Tips
- Moving the decimal in the wrong direction. To turn a decimal into a percent, multiply by 100: \(0.07=7\%\). To turn a percent into a decimal, divide by 100: \(7\%=0.07\).
- Dropping a zero in a small probability. \(3\%\) is \(3/100=0.03\), not \(0.3\). A quick check is to convert back: \(0.3\) would be \(30\%\), not \(3\%\).
- Rounding before finishing the conversion. Keep extra digits while working, convert to the requested form, and round at the end. Early rounding can affect the final digit.
- Calling a rounded value exact. If \(7/24\) is written as \(0.292\), the decimal is approximate. Keep the exact fraction when exactness matters, or use \(\approx\) to show a rounded value.
- Rounding to the wrong place. “Nearest tenth of a percent” means one digit after the decimal point in the percent. “Nearest hundredth as a decimal” means two digits after the decimal point in the decimal probability.
- Leaving off the percent symbol. \(0.225\) and \(22.5\) are not the same probability. Write \(22.5\%\), not \(22.5\), when giving the percent form.
For a clear AP response, show the conversion that matches the requested form, include the percent symbol when appropriate, and label rounded values as approximate. If helpful, check by converting the answer back to the starting form. A probability such as \(0.225\) should become \(22.5\%\), and converting \(22.5\%\) back should recover \(0.225\).
Check Your Understanding
For each question, show the conversion and pay attention to the requested form or precision.
- Convert \(3/25\) to a decimal and a percent.
- Convert \(8\%\) to a decimal. Then explain why \(0.8\) would not be the correct decimal.
- A probability is \(5/18\). Its decimal value is \(0.2777\ldots\). Round it to three decimal places and convert the unrounded value to a percent rounded to the nearest tenth of a percent.
- Convert \(0.07\) to a percent. Then convert your percent answer back to a decimal as a check.
- A context asks for a probability to the nearest hundredth of a percent. How many digits should appear after the decimal point in the percent, and should you round before or after converting?