Is it worth taking risk?

Me and my friend were on vacation in Las Vegas. We both have contradicting personalities. He is a risk taker while I am not. We were playing the Guessing Game in one of the casinos. The game is very simple. There are three boxes say A, B, C where only one of the boxes contains the prize. You need to play all three steps of the game before you claim the prize.

Step 1: You are supposed to guess the box containing the prize. Ofcourse all three boxes are equally probable. Say you chose box A.
Step 2: The game host will then open one of the other two boxes which does not have the prize. That is he wont open the box A as you have already chosen it in step 1. Hence, he will either open box B or C depending on which ever doesnt have the prize. Lets say he opened Box B.
Step 3: In this step, you can either claim the prize from the box A which you chose in first step or you can choose the other unopened box (Box C).

My friend claims, you will always increase your chances of winning if you choose a different box in step 3 rather than trusting your first instinct in step 1. But I argue it doesnt matter as both boxes have equal probability of containing the prize. Do you think taking the risk of switching the boxes increases your chances of winning?

NOTE: There are no tricks in the question. It is purely logical reasoning and the laws of probability.

[HINT: Try to think from the point of view of the game host rather than as a player.]