TITLE: Probability to find two numbers in the same group
QUESTION [5 upvotes]: The integers from 1 to 100 are randomly separated into two groups of 50 integers each. What is the probability that 37 and 89 are in the same group?
My reasoning in this exercise is to first put one number in one of the possible 100 positions and then place the second one : For the first one there is no restrictions and for the second one there are 49/99 possible position.Besides the positions of the other numbers are not important so i think that the answer is 49/99 , am i right?

REPLY [2 votes]: Your reasoning and your answer of $\frac{49}{99}$ are correct
Another less efficient approach might be to say there are ${100 \choose 50}$ equally likely different possibilities of choosing the first part, and of these ${2 \choose 2}{98 \choose 48}+{2 \choose 0}{98 \choose 50}$ have your two specific values together in either the first or the second part.  Divide the latter by the former and simplify to get the same result