Question

Problem 3: Probability of Collision

In country of Crapoptamia, a motorized vehicle’s license plate has the form ‘DLLLDDD’ where D is a digit [0,9] and L is an upper-case English letter. In positions 2 and 4, letters I, O, and Q are not allowed.

In a Math course, the instructor tells the class that, assuming that each student has only one car, there is a 99% probability that at least two students have the same first three characters on their license plates.

1. How many students are there in the class? Explain your solution.

2. One student says that his car license plate starts with ‘4TG’ and asks his classmates to raise a hand if anyone has the same first three characters. Weirdly, no one responds. Has the instructor made a mistake? Think about relevant properties of hash functions and contrast the instructor’s logic and the student’s question.

Answer 1

1)

Let's first look at the all possible different first three characters.

The total number of those are 10×23×26 =5980.

Now if there were 5981 students in the class, the instructor could say that " there is 100% probability that at least two students have same first three characters .

So the number of students are 5980× 0.99 = 5921

2) No the instructor hasn't made a mistake. He merely stated there is 99% chance that two people will have same first three characters in their vehicle's license plate. And he hasn't said anything about which particular sequence of the first three characters will be repeated.