I was playing Scrabble with relatives and came across this conundrum. At the beginning of the game, each player is to draw 7 tiles at random from a bag. Is there a difference between each player drawing 7 tiles at once and then passing the bag to the next player, versus each player drawing 1 tile and then passing the bag to the next person and having the bag passed around 7 times?
Probably not, depending on what you’re interested in. The draws are exchangeable, which means that any outcomes that only differ with respect to the order of the draws must have the same probability.
If the draws are truly random, there’s no difference. There’s no outcome that would be any more or less likely under the first method than under the second.
Rationally, no. You’re dealing with 7 unknown tiles out of the total tile pool in either case.
Players can be very superstitious, though, and can get very worked up if you break from routine, especially if they get a bad draw out of it!
Likewise, if you’re playing cards, it wouldn’t matter whether you deal the cards one at a time, or give the first player all of his cards at once—if the deck is well-shuffled, with all of the cards in a completely random order. In practice, however, you can’t count on that, so it’s generally fairer to deal the cards one at a time.
This is one of those paradoxes where the odds change if you look at the tiles, isn’t it?
Not sure what you mean (or even if you’re serious). If you look at the tiles after they’re picked, nothing changes (except the players find out earlier what they have and don’t have). Obviously if the players can look at the tiles as they’re picking, then of course things aren’t random, as the first player is probably going to have a “Q” or “Z”, and six other letters that form a word.
I meant that a player picks a tile - or all seven tiles - at random and then looks at it / them.
Looking at the tiles as they’re drawn doesn’t change the answer to the original question. Drawing 7 at a time or 1 at a time offers the same tile probabilities for your hand.
There are questions to which looking at the tiles changes the answer, but they would be fairly contrived ones in the context of Scrabble.
Then he or she could make more accurate predictions of the draws of the other players, but the overall distribution of tiles will be the same.
Then it becomes a question of semantics, not a question of math. IIf you draw a card from a deck, and don’t look, some would say that you have a 1/13 chance of having a King. Others would say that there’s a 1 probability or a 0 probability of you having a King and they just don’t know what it is yet. That’s not really a mathematical question; it’s just a fight over what it means to “have a chance” or “have a probability”. Then you have to get all philosophical about destiny and self-determination and whatnot.
From a certain perspective, yes, looking at your tiles “changes” the probabilities. But that’s only because you have more information than you had before. Before you look, you can say “I might have a G” but after you look down at a Q, you can’t say that anymore. There’s no paradox here, though.
It could make a difference if you were given a choice to continue picking tiles or pass the bag to the next person. If I drew a “good” tile that means the pool of tiles is now worse.
How is that any different from the case where you pass the bag after a prescribed number of draws? (I’m not saying that it isn’t, but it’s not immediately obvious to me what could go wrong.)
Assuming that all tiles are face-down and well-scrambled (or in an opaque bag, or whatever), you could even have a person draw tiles one at a time, look at each one as he draws it, and then decide whether to pass the bag, and still not be able to give himself an advantage over any other player.
I got into a similar argument with someone I was playing Spades with. (In Spades, all 52 cards are dealt to 4 players.) A child was playing, and she had difficulty reaching across the table repeatedly while doing a round-robin deal. So she instead dealt 13 to me, 13 to the next player, 13 to the complainer, and kept the rest.
When the 3rd player complained about this not being random, I asked if our young friend’s dealing had messed up his fixing the deck. He shut up after that.
The observation of the tiles effectively eliminates some permutations from the sample space which makes it natural and logical to consider conditional probabilities for the unobserved tiles. However, this has no effect whatsoever on the initial distribution of the tiles before the game starts. Nor does the dealing method if the tiles are shuffled randomly.
I’m always a defender of taking just one tile (or card) at a time. As others have said, a perfectly random set means that the order of draw doesn’t matter.
However, there’s often some very shoddy shuffling going on. In my family, the favorite way to mix Scrabble pieces wasn’t effective - we slid tiles off the board, turned them face down and just shook the box lid back and forth. The tiles went back and forth in the box lid, but adjacent tiles stayed adjacent. Picking a handful of adjacent tiles gave you a decent chance of getting a chunk of a word that came off the board from the previous game. In fact, drawing from the left gave you a good chance of getting letters from the left side of the board. The same analogy holds for cards - if lousy shuffling leaves 4 Aces together, at least taking one card at a time will split them up.
As a kid, I did a little informal checking on my own to see how well/badly things got mixed, and verified it by turning all the tiles/cards face up after shuffling. There were definitely a lot of non-random sequences of tiles/cards.
(I did a lot of stuff like that as a kid. When I wanted to verify whether smooth edges on a 10-sided die made it more or less random than sharp edges, I rolled each 1,000 times. No difference between the dice, for those who are interested.)
I don’t understand this. If I draw 7 tiles, then that means that there are only 93 tiles left for the next person. If I draw 1 tile, there are 99 tiles left for the next person.
dracoi, you rocked as a little kid. I just wanted to get that out there.
Oh, and I wanted to add, bad shuffling doesn’t really randomise very well. Which is why Magic the Gathering players tend to manually “randomise” their deck by interspersing their land cards (think about 20 cards that all do the same thing) in the deck before shuffling. Sure, it’s not really random, but you don’t want to shuffle badly and end up with a hand full of lands, or a hand with no lands at all.
I’m not a statistician, but it seems to me the OP’s two methods DOES make a difference. Doesn’t drawing seven tiles at once better my chances for drawing the one “X”? If I get it, now I’ve increased the odds for consecutive players that they WON’T get the “X” (i.e., a 100% chance they won’t). If I don’t get it on my first turn, now I’ve increased the odds the next person will because there are now seven less tiles in the remaining total population.
Say it were cards and I add one joker to the deck. I draw seven cards at once leaving the next guy with a 1 in (53-7) chance vs. a 1 in (53-1) chance of drawing the sole joker.