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Probability Card Puzzle?
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
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