What are the odds?

[Dave Potts]

If you were to deal out 11 cards at random from a full deck of 52 cards, what would the odds be of all 11 cards being of the same suit, and how in the name of Alan d'you work that out? Ta.

[Sylday]

I make it about 780 million to one against the all eleven cards will be of a particular suit. The way I work this out is that the probability that the first card will be of a particular suit is 13 in 52, or 1 in 4, which can be represented as 0.25. On the second card the probablility is 12 in 51, or .235; and so on until before dealing the 11th card the probability will be 3 in 42 or .071. You then multiply together all these probability fractions, which gives you 1282357 with 8 noughts in front; then by dividing this fraction into one, you get the figure of 779,814,045, gives you the odds against it all happening.
Now someone, tell me where I went wrong!

[Dave Potts]

Lest anyone be wondering why I'm asking, this exact situation came up in a poker game last week where 3 players all had 2 hearts as their 'hole cards' and the 'board' was five hearts. We beat up the dealer.

[rja211077]

yeah Sylday's got the right idea but since Dave didn't specify a suit but merely stipulated that all 11 be the same, I would have left off the term associated with the first card, so the correct answer is 4 x Sylday's answer.

[Shrek2]

Whats the probability that the dealer set it up ???

[bernardo]

Sylday has got the correct answer mathematically, but in a real life situation like the one encountered by Dave Potts, the flaw lies in the use of the word "random". It is highly likely that the cards were not shuffled enough properly before the dealing was done. Even someone who is honestly shuffling a deck of cards, and who is not deliberately trying to fix things, could quite easily not do it thoroughly enough. It depends on how the shuffling is done (i.e. the physical method of moving the cards relative to each other) and how much. Coincidences of this type are reported (honestly or otherwise) far more often than would happen by chance, so it should be assumed that the cards were not properly randomised (in the sense which weould be recognised by a mathematician) in the first place.

Incidentally, it is far more often that people claim to have had 4 players each receiving a whole suit of 13 cards than that only one out of four players has done so - even though the latter is overwhelmingly more likely in a random situation.

[Dave Potts]

Couldn't agree more with bernardo regarding the inefficacy of a human shuffle in producing a wholly random deal, although most people would probably maintain that our cards are generally pretty well shuffled i.e. the cards are spread face down on the table and given the old 'wipe-on, wipe-off' treatment before being riffle-shuffled a good few times and then cut. In the case of our games, one player shuffles, another player cuts and another player deals so it's doubtful whether there was any deliberate artifice on the part of the dealer. In the case of the deal one card at a time was dealt in rotation to each player (there were 5 at this table, but 2 folded) then a card was discarded (or 'burned) prior to 3 community cards being dealt. Another discard, another community card, then another discard followed by another community card. There were no hearts among the 3 discards or the 4 'mucked' cards. I suppose you can never get a truly random shuffle but I was simply curious about the stats assuming the randomness of the deal to be a given, so my thanks to Sylday for the mental gymnastics and to everyone else for their contributions!

[Shrek2]

from looking at Dave's response, it seems that it was pure chance that the cards fell the way they did. Happens once in a blue moon.

[berty2001]

It could be argued that it is one in four as each deal of the card poroduces its own 'odds', and the chance of chosing a certain suit each time is one in four. Or one in 52 as the cahnces of pulling out a certain card is one in 52. I think some people aregue that probability should be measure this way as what has happened before has no bearing on what is to happen after in this situation. Of course the odds would get higher as you went along for each suit as there would be less of that suit and more of the others left in the pack, so it would be slightly more than one in four.

[tomd]

These all only apply if the deck is fully shuffled.

[Dazzler]

The probability is 100% That's the only way this question would be asked, and it has been. When co-incidences happen, we notice them and we mentally discard all the times that nothing unusual happens, making the unusual appear more common. A bit like only having memories of very good times and very bad times, if we remebered all the "average" times we'd soon run out of storage capacity. Not the answer to the question I know, but I was feeling creative, sorry