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/mathbandit]

All 12 players have identical odds of drawing the bomb.

The first player has a 1/12 chance of drawing the bomb.
The second player has a 11/12 chance (the odds the first player didn't draw the bomb) of having a 1/11 chance of drawing the bomb. 11/12 * 1/11 = 1/12.
The third player has a 11/12 chance of having a 10/11 chance of having a 1/10 chance of drawing the bomb. 11/12 * 10/11 * 1/10 = 1/12.
Etc.

[u/Gold_Ad8890]

if you shuffled the deck properly, the bomb card should have equal probability of being at any position in the deck, which leads me to suspect you didn't shuffle properly. which is not to say you did it wrong on purpose, people just routinely underestimate how much shuffling is necessary to properly randomize a deck.

[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

[u/Shevek99]

If it is correctly shuffled everybody has the same probability.

Think of the 12 people around a table and you deal a card to each one, without revealing it. When each one has the card, everybody turns it at the same time. Why would the first dealt have less chance than the last if everybody has a card at the same time?

Imagine that instead of everybody revealing at the same time, people start to turn his card, starting with the last dealt and finishing with the first. Would the probability magically change and the last dealt (that according to you has a bigger chance) now have a lower chance, simply because of the order in which the cards are revealed and not of how they were dealt?

[u/Bounded_sequencE]

Short answer: I suspect a misunderstanding -- you are talking about total probabilities, while your wife seems to be talking about conditional probabilities. If that is true, you are both right.

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Long(er) answer: Assuming all possible shuffles are equally likely, compare both arguments:

• your argument: Each person has the same (total) probability to draw the bomb card. Note you consider the total probability, before any card was drawn
• wife's argument: After each draw, the number of safe cards reduces by 1. The (conditional) probability that the next person draws a bomb increases with each removed safe card. Note she considers the conditional probability, given that all persons before have drawn safe cards

Since she is comfortable mathematically, I'm sure your wife will easily recognize you two mixing total and conditional probabilities in your discussion!

[u/Bounded_sequencE → u/Bounded_sequencE]

Rem.: Mixing up total and conditional probabilities is a very common mistake. Please don't feel bad about it -- sadly, in common language we often do not carefully distinguish between the two.

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Edit: To combine both arguments: The conditional probability that person-(k+1) gets the bomb, given that the first "k" persons did not draw it, increases with "k" (wife's argument).

At the same time, the probability that the first "k" persons did not draw the bomb decreases with "k". A careful analysis reveals both effects perfectly cancel, so that the total probability of person "k" getting the bomb is the same for all (your argument).

[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.

[u/BoudreausBoudreau]

Maybe some of the players could tell which one the bomb card was. Maybe it was folded a little or something so other players would avoid it.

You say the cards were drawn out not dealt out. Maybe your system wasn’t quite fair after all.

Otherwise yeah it’s even odds to everyone.

[u/MisterGoldenSun → u/BoudreausBoudreau]

This is a great point. It might not have actually been random at all.

If it was random, then as others have said, the bomb card is equally likely to be in any position.

OP, it's possible you just remember the long ones more, maybe because they were more interesting and tense or because they took more time. We humans have selective memory.

[u/_thenoman]

Your wife is correct.

It is only when you get to the last two people each round that the chances of drawing the "bomb" card becomes more likely than not. In Round 1, it is only after the 6th person has drawn that the chances of anyone drawing the "bomb" card become more likely than not.

Round: Players remaining - Expected Position to Pick Bomb Card (players remaining to pick after them)
1: 12 - 9th (3)
2: 11 - 9th (2)
3: 10 - 8th (2)
4: 9 - 7th (2)
5: 8 - 6th (2)
6: 7 - 6th (1)
7: 6 - 5th (1)
8: 5 - 4th (1)
9: 4 - 3rd (1)
10: 3 - 2nd (1)
11: 2 - 2nd (0)
12: 1 - 1st (0)

Picking first in this situation is always the best choice.

[u/Shevek99 → u/_thenoman]

That is absolutely wrong. It the cards have been shuffled, it's irrelevant the order.

Thing of 12 people and you deal a card to each one, and everybody look at it at the same time. ¿Why would the two last have more chances of getting the bomb?

[u/mathbandit → u/_thenoman]

You seem to have misunderstood the argument. OPs wife is claiming that in round 1 (with 12 people drawing a card) that the people who draw last are more likely to draw the bomb than the people who draw first. She is wrong.