Odds of 2 people answering exactly the same on a 1000 question multiple choice test What are the odds of 2 people answering the same on a 1,000 question multiple choice test with approximately 50,000 people taking the test ?
The test was about random things. Life, the world, likes, dislikes, etc. They had four options to choose from  
I have a friend that claims that her and someone else took a 1,000 question test and her and another person got a perfect match. They became friends and he verifies the story also. The next closest was a 85% match, with most people scoring having a match around the 70-75% range. 
So, we all want to know what are the odds of that happening.
Thank you.  
 A: In the interest of giving some kind of bound, let's assume everyone was answering the questions randomly. As others have noted, this assumption gives a lower bound on the probability. If the choices are about equally-likely to be selected, then the bound will be close to the truth.
Let's ask the dual question first: What is the probability that no two test-takers submitted the same answers? 
Since each question has four options and there are $1,000$ questions, there are a total of $4^{1000}$ different ways to complete the test. We can select $50,000$ distinct submissions from this collection in $(4^{1000})_{50,000}$ ways (the falling factorial). The total number of possible submissions is $(4^{1000})^{50,000}$. Finally, the probability that no two tests had exactly the same answers is
$$
\frac{(4^{1000})_{50,000}}{(4^{1000})^{50,000}}, 
$$
which is quite close to $1$.
Returning to the original question, the probability that at least two answered the same is 
$$
1 - \frac{(4^{1000})_{50,000}}{(4^{1000})^{50,000}}, 
$$
which is quite close to $0$.
