Problem 5: Probability of Collision In the university of Crapoptamia, every student can design their own student ID. Here is the rules for getting the student ID: 1. The rst character should be an uppercase English letter. Students can randomly pick one letter. 2. The following two digits should be the month of student's birthday. 3. The last four characters are 4 digits chosen by each student. libr example, 30551133 is an valid ID for a student who was born in May. We assmne that each student will randomly choose the rst English letter and the four digits. [n a Computer Science class, the instructor tells the class that there is at least a 99% possibility that two or more students share the same rst four characters in their student ID. [a] At least how many students are there in the class? Explain your solution. (Assume that the distribution of the birth months is uniform.) {b} One student says that the rst four characters are 'Al'' and asks his classmates to raise the hand if someone has the same rst four characters. However, 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. Solution: {a} Let Pg be the probability of no collision. As the probability of collision is at least 99%, we have: P, <_: let n be the distinct values in rst four characters. english letter should range of z following digits and fourth character will randomly chosen from so we have> 170.02 2 Thus, there are at least 171 students in the class