Question

R CODE:

1. A bag contains n distinct (all different) numbers: (x1; x2; : : : ; xn) . Per-

son A draws a number at random, while person B draws a number from

the remaining choices. What is the exact probability that A's number is

greater than B's number? Explain your reasoning in detail.

Answer 1

The exact probability is 1/2. We can derive this mathematically, but before that let me give you a more intuitive explanation.

Since both the numbers are always distinct, let A choose x_i and B choose x_j, x_i =/= x_j
Then either xi<xj or xi>xj.
Without loss of generality, assume A<B. Then, swap the choices of A and B and we get A>B. This can be done for every choice of A and B. Thus there are exactly equal number of outcomes for both the cases. Thus the probability of A's number being greater than B's number is exactly 1/2.

A more mathematical approach is given below.

You can also run an R program to simulate this scenario and see the result approaching 1/2
If you can't write the code yourself, follow up in comments.