Question

Players A and B each have a well shuffled standard pack of cards, with no jokers. The players deal their cards one at a time, from the top of the deck, checking for an exact match. Player A wins if, once the packs are fully dealt, no matches are found. Player B wins if at least one match occurs. What is the probability that player A wins?

Answer 1

you gotta be kidding.
this is a real pain

Answer 2

I am not kidding.

Answer 3

No googling allowed!

Answer 4

oh it is a quiz!

Answer 5

haha how are you gonna enforce that...that tells me this question is out there somewhere huh?
:)

Answer 6

OUI!YES!

Answer 7

can i write a computer program to simulate the game, then determine the probability

Answer 8

Well you can do that.

Answer 9

But you can also use your brain instead of a CPU! :)

Answer 10

yes the answer is out there, as you might imagine it is a standard problem, and the solution is cute

Answer 11

is the probability that on any given turn they have a match 1/52 ?

Answer 12

well my guess is
\[(\frac{51}{52})^{52} = 0.364\]

Answer 13

Use the inclusion - exclusion principle.You'll get a surprising result!

Answer 14

For any reasonably large number of cards, say, 10 or more, the probability that A wins is almost independent of the number of cards in the decks!

Answer 15

The probability that A wins is d(52) / 52! = 1 − 1/1! + 1/2! − 1/3! + ... + 1/52!

Answer 16

or,1/e

Answer 17

ah ok, well its time to get off OS
oh solution thanks...hey i was close

Answer 18

Yeah!You were!

Answer 19

http://www.wolframalpha.com/input/?i=1%2Fe&t=crmtb01

Answer 20

just wanted to add something i noticed
\[\lim_{n \rightarrow \infty}(\frac{n-1}{n})^{n} = \frac{1}{e}\]
so 1/e is a good approximation for large n
but the exact solution for deck of 52 would be (51/52)^52
no?

Answer 21

@dumbcow i think your answer, assuming it is right and it certainly looks right, is much simpler than the answer given in the nicks puzzles with all the "inclusion/exclusion" summing over stuff business
i am wondering why, because your solution sure looks correct and straightforward, or am i missing something?

Answer 22

i don't know much about inclusion/exclusion honestly
i wasn't really trying, but when he said solution was 1/e it made me pretty sure my answer is the correct one for a deck of 52 cards

logic: chance of a specific card is drawn is 1/52 for each player
probability they match = 1/52 * 1/52 = 1/52^2
there are 52 different cards so probability any given card matches = 52/52^2 = 1/52
probability they don't match = 51/52
for 52 trials --> (51/52)^52