Question 9·Hard·Probability and Conditional Probability
The table below shows the distribution of U.S. states according to whether they have a state-level sales tax and a state-level income tax.
| State sales tax | No state sales tax | |
|---|---|---|
| State income tax | 39 | 4 |
| No state income tax | 6 | 1 |
To the nearest tenth of a percent, what percent of states with a state-level sales tax do not have a state-level income tax?
For two-way tables and conditional probability questions, first underline the condition phrase like "of states with a state-level sales tax" to remind yourself that only that row or column is your total. Add the relevant cells in that row or column to get the denominator, pick the cell that matches the event described in the question as the numerator, form the fraction (numerator over conditional total), and then convert the fraction to a percent. This approach prevents the common mistake of dividing by the total number of all items in the table instead of the conditional group.
Hints
Focus on the correct denominator
The phrase "of states with a state-level sales tax" tells you which group you should be using as the total. Which entries in the table belong to this group?
Add the correct cells from the table
Look only at the State sales tax column. Add the two numbers in this column to find how many states have a state-level sales tax in total.
Form the correct fraction
From the states that have a state-level sales tax, identify how many do not have a state-level income tax. Put that number over your total from the previous hint to form a fraction.
Convert the fraction to a percent
Once you have the fraction, turn it into a decimal (by dividing) and then multiply by 100 to write it as a percent. Round your answer to the nearest tenth of a percent.
Desmos Guide
Compute the conditional percent in Desmos
In Desmos, type the expression 6/(39+6)*100 and press Enter. Then read the numeric output and round it to the nearest tenth to get the percent value.
Step-by-step Explanation
Identify what group the percent is taken "of"
The question asks: "what percent of states with a state-level sales tax do not have a state-level income tax?"
This means:
- We are only looking within the group of states that have a state-level sales tax.
- Our denominator (the "total" in the fraction) must be the number of states with a state-level sales tax.
Find the total number of states with a state-level sales tax
Look at the State sales tax column in the table:
- With state income tax: 39
- With no state income tax: 6
So, the total number of states with a state-level sales tax is:
This 45 is the denominator for our fraction.
Find how many of those do not have a state income tax
Still in the State sales tax column, we want states that do not have a state income tax.
From the table, that number is 6.
So, within the 45 states that have a sales tax, 6 of them do not have an income tax. Our fraction is:
Simplify the fraction and convert to a percent
First, simplify :
Now convert to a decimal:
Convert this decimal to a percent by multiplying by 100:
To the nearest tenth of a percent, the answer is 13.3%, which corresponds to choice C.