I shuffle a deck of cards. Each card appears there only once. Now, I need to pick a card randomly. I can do this in two ways:
- literally, pick a card somewhere in the beginning/middle/end of the deck randomly.
- always pick the first card
Context: for many games, you start by picking a random card to determine various things: how to start, who starts etc.
My friend claims that these two ways yield the same result, essentially they are the same i.e. each card has an equal probability of being at the very first place in the deck since we are shuffling randomly.
I claim that:
1) it COULD only be the case IF we could ensure "universally equal" shuffling, meaning that based on the person's shuffling, some cards might never make it to the top,
2) since we do not have such "universally equal shuffling", the probabilities of the cards being picked are 1 for the first card, and 0 for all others; while if you randomly pick a card (way #1) the probabilities are indeed all equal to $1/52$ for each card (assuming there are 52 cards in the deck).
Where is the truth?