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/QuickKiran]

Assuming the bomb card is distributed uniformly (fancy math speak for "the deck is shuffled fairly"), the advantage people at the end have (more likely to not have to choose because someone before them draws the bomb) and the advantage people at the beginning have (more safe cards) balance out perfectly. People at the beginning, middle, and end all have the exact same odds of drawing the bomb card before the round begins. 

If you'd like to see this mathematically, if you're in seat s, what are the odds of drawing the bomb? On your turn, there are s cards left, so 1 in s. But you also need to make it to your turn. That means the first person doesn't draw the bomb (11 in 12, if there are twelve players, n-1 in n for a more general n players), then the second player also doesn't (10 in 11, or n-2 in n-1), and so on for everyone before you. Then we multiply these probabilities together: 11/12 * 10/11 * 9/10 * ... * s/(s+1) * 1/s Everything cancels and you're left with 1/12 regardless of what s is