Question 11·Medium·Probability and Conditional Probability
United States Presidents from 1789 to 2015
| Age at inauguration (years) | Number of presidents |
|---|---|
| 40–44 | 2 |
| 45–49 | 7 |
| 50–54 | 13 |
| 55–59 | 11 |
| 60–64 | 7 |
| 65–69 | 3 |
The table above gives the number of United States presidents from 1789 to 2015 whose age, at the time they first took office, falls within each interval listed.
Of those presidents who were at least 50 years old when they first took office, what fraction were at least 60 years old?
For conditional fractions and probabilities on the SAT, always start by identifying the restricted group named after phrases like "of those" or "among"—this group provides your denominator. Sum only the rows (or data points) that belong to that group, then separately sum the subgroup the question is asking about (the numerator). Finally, form the fraction numerator/denominator and compare it directly with the answer choices, being careful not to use the total from the entire table unless the question explicitly asks for it.
Hints
Focus on the phrase "of those"
When a question says "Of those presidents who were at least 50 years old," it means you should only look at that group for your total (denominator). Which rows of the table are ages 50 and above?
Find the denominator first
Add up the number of presidents in all the age intervals that are 50 or older. That total will be the bottom of your fraction.
Then find the numerator
Now count only the presidents in the 60–64 and 65–69 rows. That total will go on top of your fraction, over the total you just found.
Build the fraction
Use the count from the 60+ group as the numerator and the count from the 50+ group as the denominator. Do not use the total number of presidents from the whole table.
Desmos Guide
Compute the total number of presidents at least 50
In Desmos, type 13+11+7+3 to confirm the total number of presidents who were at least 50 years old. Note the result; this is your denominator.
Compute the number of presidents at least 60
Next, type 7+3 to confirm how many presidents were at least 60 years old. This will be your numerator.
Form and evaluate the fraction
In Desmos, type (7+3)/(13+11+7+3) to see the exact fraction (and decimal) representing the fraction of presidents who were at least 60 among those who were at least 50. Match this fraction to the correct answer choice.
Step-by-step Explanation
Identify the relevant age groups
"At least 50 years old" includes all presidents in the age intervals 50–54, 55–59, 60–64, and 65–69.
"At least 60 years old" includes the presidents in 60–64 and 65–69.
Find the total number who were at least 50 (the denominator)
Add the numbers in the 50+ rows:
- 50–54: 13
- 55–59: 11
- 60–64: 7
- 65–69: 3
So the total number of presidents who were at least 50 is:
.
This 34 will be the denominator of the fraction.
Find how many of those were at least 60 (the numerator)
Now look only at the 60+ rows:
- 60–64: 7
- 65–69: 3
Add them:
.
This 10 is the number of presidents who were at least 60 years old when they took office.
Form the conditional fraction
The question asks: Of those who were at least 50, what fraction were at least 60?
So the fraction is
That corresponds to answer choice B) 10/34.