Math problem: passing the U.S. Citizenship test the lazy way

There’s not much to prepare for in the U.S. citizenship test.

Perhaps the hardest part is a section where they ask history and government questions.

The interesting thing is how this section is structured. They ask a sample of questions of which one must answer a majority correctly. Here is how the U.S. government describes the test:

There are 100 civics questions on the naturalization test. During your naturalization interview, you will be asked up to 10 questions from the list of 100 questions. You must answer correctly at least six (6) of the 10 questions to pass the civics test.

from USCIS website

I should note that the U.S. government provides all 100 questions and answers to applicants, so all exam questions are known in advance.

While it might be safer to prepare for all 100 questions, the test structure means one can obviously study less.

The question is: what’s the minimum effort required?

It depends on whether you want to be sure of passing the test, or if you’re comfortable on having a high chance of doing so.

Here are a few questions that come to mind:

1. How many questions must be prepared (memorized) to guarantee passing the test?
2. How many are needed to achieve an 80 percent chance of passing the test?
3. If a test-taker only prepares for 50 questions, what are his chances of passing?
4. How many questions must someone prepare for to yield at least a 50 percent chance of passing the test?

Answers to these questions are below.

1. How many questions must be prepared (memorized) to guarantee passing the test?
This question is easier to answer than the following ones.

Since a test-taker can miss 4 questions without penalty, it follows that one can be ignorant of 4 questions without harm.

Thus, one must study 96 questions to be sure of passing the test.

(If you were to study only 95, there is a chance you’ll be asked the 5 questions you don’t know and will fail.)

The method to solve the remaining questions
We need to figure out the following: if you prepare for x questions, what is the chance you’ll get 6 or more questions correct out of 10?

The math is described in the wonderful blog Math Goes Pop! (this is definitely a site to subscribe to).

In one post, the general method is described. In a follow-up, the problem of the U.S. citizenship test is examined.

Those articles describe all the nitty gritty details so I will not do so here.

I will instead describe the general idea. We want a way of calculating the odds given that we study a certain number of questions, and we sample a subset from the total population.

One way to think about this is using the classic model of picking white and black balls from an urn. We can think about the 100 questions as 100 balls in an urn. The number of questions we prepare for are “white” balls, the number of questions we don’t know are “black” balls. The question then becomes: if we pick 10 balls from the urn (without replacement), what is the chance we get at least 6 white balls (we know at least 6 questions).

Using this analogy, it is easier to see the test preparation is described by the hypergeometric distribution. (It is not a binomial distribution because the sample is drawn without replacement).

The idea is to then calculate the associated probabilities for getting 6, 7, 8, 9 and 10 questions correct based on preparing for anywhere from 6 to 96 questions (less than 6 and you’re sure to fail, more than 96 and you’re sure to succeed). Then sum up those probabilities to see the chance of getting at least 6 questions correct.

Here is what a graph looks like

See the end of this article for the detailed table of calculations.

2. What about 80 percent sure?
Using the table at the end, we can read you need to prepare for 67 questions to have at least an 80 percent chance.

That means you only have to study about 2/3 of the questions to have a very good shot — 8 out of 10 times — of passing.

3. If a test-taker only prepares for 50 questions, what are his chances of passing?
Again reading the table, the answer is 37 percent.

It is interesting to note that you can study half the questions but have only about a third chance of passing the test.

This means people unprepared are unlikely to pass just by simple chance.

4. How many questions must someone prepare for to yield a 50 percent chance of passing the test?
From the table, you would have to memorize the answers for 55 questions to exceed 50 percent.

Learning just 5 extra questions boosts the chance of succeeding by more than 10 percent. Evidently studying is most beneficial for test-takers who start out with an average level of knowledge in civics.

Table of calculations