# What is the probability when using two separate card decks that the same card will be turned at the same time

Ok I think this is very easy. In fact I think it may be the following equation:

P(A) = 1/52
P(B) = 1/52
P(A and B) = P(A) * P(B) = 1/2704


However it doesn't feel right. If you play this out for real the odds seem a lot better than 1 out of 2704.

Could someone enlighten me?

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What are A and B? – Chris Eagle Sep 29 '12 at 14:22
P(A) represent the odds of a 52 card deck same for P(B) – HGPB Sep 29 '12 at 14:25
In a probability question, the experiment has to be described in detail, else there can be several different interpretations, with several quite different numerical answers. – André Nicolas Sep 29 '12 at 15:21
@AndréNicolas understood. But doesn't the title describe the problem quite well? – HGPB Sep 29 '12 at 16:09
@Haraldo: Not really. Are the decks kept separate, with Alice and Bob turning up the top card? Do they only do turn up the top cards, or do they go through the entire deck, as in the children's game of War? Or are the decks shuffled together and the top two cards are turned up? Or $\dots$. – André Nicolas Sep 29 '12 at 16:13

If $A$ and $B$ represent the cards turned over in the first and second decks, then $$\begin{eqnarray} P[A=x] = P[B=x] &=& \frac{1}{52} \\ P[A=x\wedge B=x]&=&\frac{1}{52^2} \end{eqnarray}$$ for any particular card $x$. The probability that $A$ and $B$ are equal to each other is then $$P[A=B]=\sum_{x}P[A=x\wedge B=x]=52\times\frac{1}{52^2}=\frac{1}{52}.$$

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I would vote you up but I don't have enough rep! – HGPB Sep 29 '12 at 14:40
Does the dot represent something after the 1/52 answer? – HGPB Sep 29 '12 at 16:13
The end of a sentence :). – mjqxxxx Sep 29 '12 at 16:23
lol - I thought it might be some maths notation ;) – HGPB Sep 29 '12 at 16:31

No. The first card doesn't matter what, it can be 52 out of 52, and only $B$ is restricted then. So, it is $\displaystyle\frac 1{52}$.

What you answered is the probability that both $A$ and $B$ are the ace of spades, for example.

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not really, it should be $1\over 103$, as two deck is combined so there will be 103 card left. Your answer is fine with 2 separate deck – Mathematics Sep 29 '12 at 14:26
Your second point is what I am actualy looking for... – HGPB Sep 29 '12 at 14:27
Oh, I understood the 2 decks are separate.. – Berci Sep 29 '12 at 14:29
@Mathematics Thanks, so its a 1/103 chance of both turning up an Ace of spades? – HGPB Sep 29 '12 at 14:30
@Berci Yes the decks are seperate. Two people turning cards over. What are the odds of them both turning an Ace of spades at the same time. – HGPB Sep 29 '12 at 14:31