# probabiliy of equal cards

• Jun 3rd 2012, 02:12 PM
billobillo
probabiliy of equal cards
If we have two decks of 52 cards each that are shuffled randomly.And if the positions of cards for each deck are numbered from 1 to 52.
What is the probability that these two decks contain the same card at the same position.
• Jun 3rd 2012, 02:55 PM
Plato
Re: probabiliy of equal cards
Quote:

Originally Posted by billobillo
If we have two decks of 52 cards each that are shuffled randomly.And if the positions of cards for each deck are numbered from 1 to 52. What is the probability that these two decks contain the same card at the same position.

If a standard decks is well shuffled what is the probability that no card is in its original position?
• Jun 3rd 2012, 03:50 PM
billobillo
Re: probabiliy of equal cards
Quote:

Originally Posted by Plato
If a standard decks is well shuffled what is the probability that no card is in its original position?

Well I guess your question is the same of mine, because having a non shuffled and a shuffled deck is like having two different decks.
• Jun 3rd 2012, 04:00 PM
Plato
Re: probabiliy of equal cards
Quote:

Originally Posted by billobillo
Well I guess your question is the same of mine, because having a non shuffled and a shuffled deck is like having two different decks.

That is correct. So if $p$ is the probability of not having any card in its correct place then $1-p$ is the probability of at least one card in its correct place.

Do you know about derangements ?
• Jun 3rd 2012, 04:32 PM
billobillo
Re: probabiliy of equal cards
Quote:

Originally Posted by Plato
That is correct. So if $p$ is the probability of not having any card in its correct place then $1-p$ is the probability of at least one card in its correct place.

Do you know about derangements ?

Actually I'm not a math specialist, I tried to solve it in a easy way and I got p=1.
A card in position 1 in deck 1 has probability 1/52 to be equal to a card in position 1 in deck 2.
And since we have 52 cards, then the probability to get two equal cards at same position is 52 * 1/52 = 1
Could it be p=1?
• Jun 3rd 2012, 04:41 PM
Plato
Re: probabiliy of equal cards
Quote:

Originally Posted by billobillo
Actually I'm not a math specialist, I tried to solve it in a easy way and I got p=1.
A card in position 1 in deck 1 has probability 1/52 to be equal to a card in position 1 in deck 2.
And since we have 52 cards, then the probability to get two equal cards at same position is 52 * 1/52 = 1
Could it be p=1?

You cannot work this question with out derangements. As far as I know there is no other way.
Why were you given this problem to do?
• Jun 3rd 2012, 05:11 PM
bjhopper
Re: probabiliy of equal cards
Two decks are shuffled. The first is the reference deck and I interpret that the question is what is the P that the second deck will have the exact same order as the first. I would answer 52 factorial is the number of possible events.P= 1/8*10^67
• Jun 3rd 2012, 06:20 PM
Plato
Re: probabiliy of equal cards
Quote:

Originally Posted by bjhopper
Two decks are shuffled. The first is the reference deck and I interpret that the question is what is the P that the second deck will have the exact same order as the first. I would answer 52 factorial is the number of possible events.P= 1/8*10^67

You have got to be kidding!
• Jun 4th 2012, 12:13 AM
billobillo
Re: probabiliy of equal cards
Quote:

Originally Posted by Plato
You cannot work this question with out derangements. As far as I know there is no other way.