Probability Card Puzzle?!


Question: You can read puzzles like this at http://www.mindchallenger.com New puzzles and answers are posted at the beginning of each month.

This one many have gotten wrong:
Two identical packs of 52 cards A and B are shuffled thoroughly. One card is picked from A and shuffled with B. The top card from pack A is turned up. If this is the Queen of Hearts, what are the chances that the top card in B will be the King of Hearts?

HINT: the answer is NOT 1/51, 1/52 or 1/53.

Answers: You can read puzzles like this at http://www.mindchallenger.com New puzzles and answers are posted at the beginning of each month.

This one many have gotten wrong:
Two identical packs of 52 cards A and B are shuffled thoroughly. One card is picked from A and shuffled with B. The top card from pack A is turned up. If this is the Queen of Hearts, what are the chances that the top card in B will be the King of Hearts?

HINT: the answer is NOT 1/51, 1/52 or 1/53.
We know that the King of Hearts from deck A could have been transfered to deck B. We will use the law of total probability to find the probability that the card from B shown is the King of Hearts.

Use The Law of Total Probability

For a set of events A1, A2, A3, ... , An where the Ai's are mutually exclusive and exhaustive events and for any other event B

P(B)
= P(B and A1) + P(B and A2) + ... + P(B and An)
= P(B | A1) * P(A1) + P(B | A2) * P(A2) + ... + P(B | An) * P(An)

Here we have:

There is an abuse of notation as "transfer" = "transfer | Q of Hearts shown from A"

P( Draw King of Hearts) =

P( Draw King of Hearts | transfered K of H) * P(transfer) +
P( Draw King of Hearts | did not transfer K of H) * P(non transfer)

= 2/53 * 1/51 + 1/53 * 50/51
= 52 / 2703
= 0.01923788
I am sorry. You are completely correct. The answer is 52/2703. I misread my own puzzle. There is a 1/51 chance that the King was transfered over, and not 1/52 as I wrote. This is because we turned up a card that was not the King on deck A. Report It

Other Answers (4)

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  • Ory's Avatar by Ory
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    February 18, 2007
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    • The only answer I can think of is 1/52 - but you already said that was wrong........
    2/53
    2/53's because their might possibly be 2 king of hearts in pile B!!!
    1/52 chance queen is taken away from packA.
    so 51/52 shes not.

    so its 51/52 x 1/51 that the top card is a queen of hearts.


    King at the top COULD be 1/53, OR 2/53.

    Meaning (51/2652 x 1/53) + (51/2652 x 2/53) =

    51/140556 + 102/140556 =

    153/140556



    Am i right, or am i just confused? xD

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