A student and a professor each choose a number between 1 and 8 (1 and 8 are both possible choices). what is the probability that the two choose the same number

Method 1: The student chooses a number x. The professor has 8 possible numbers to choose from, but only one choice is the number x. Therefore, the probability that they both choose the same number is 1/8. Method 2: There are 8*8 = 64 possible pairs of choices that the student and professor can make. Eight of these choices are the same number for the student and the professor, namely (1,1), (2,2), ... (8,8). Therefore the probability that they both choose the same number is 8/64 = 1/8.