A Probability Problem

[u/CodyRogers10]

My wife and I did a fantasy football draft last night and we determined the draft order with a card game. There were 12 players, and every turn we would all draw cards until one person drew the one “bomb card.” Whoever drew that card was “out” and drafting at that position. (Ex: the person that drew the bomb card in round 1 drafted 12th. The person that drew the bomb card in round 2 drafted 11th, and so on) We would remove one of the safe cards, and draw until one person was left.

Afterward, my wife and I were talking about the game and I commented how odd it was that it was coming down to the last one or two people every turn. And she — who is incredibly smart and fantastic with numbers and works with them all day — said that it actually makes sense because the people drawing first are more likely to draw a safe card.

But I argued that they are more likely to have someone ahead of them draw the bomb card and that it feels like the people drawing in the middle of the group — who have neither benefit of having the best odds of drawing a safe cards nor the highest amount of people drawing ahead of them — have the highest chance of drawing the bomb card.

My wife is so much smarter than me in just about every way I can imagine and certainly with math problems, so it wouldn’t surprise me at all to be told that I am wrong here… but am I?

Edit to answer some of the questions that have been asked: I shuffled the deck before every round, and spread out all the cards so that each person could which ever one they wanted.
The back of all the cards looked the same.
Whoever drew first would pick the card, show the group what card it was, and then the next person would do the same.

[u/get_to_ele]

My first question is “did you (a) make the cards from paper or did you (b) actually take real printed playing cards and just pick one to be the bomb card? E.g. joker is bomb card.” Because if you (a) made your own cards, I bet it’s easy to tell which one is the bomb card or at least to figure out some of the safe cards. Also home made cards are garbage and hard to shuffle well. Doing a game like this, you need to take a real printed commercial cards and shuffle.

Second question is “Exactly how late did the bombs get drawn? You may perceive it as ‘late’ but that’s just your impression. Even one very late bomb can skew your whole perception of what the ‘mean turns to bomb’ actually was. AND you may just have a poor perception of what randomness can produce. Technically EVERY possible sequence of bomb draws is equally likely, including drawing bomb first in all 11 rounds.”

Bottom line is that any timing for the bombs is equally likely, if you shuffle properly and players can’t cheat.

So yes, both your wife and you are wrong on this one.