Probability (involving conditioning)

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SUMMARY

The discussion centers on calculating the probability of drawing an ace from a shuffled half-deck of cards after an ace has been transferred from one half to another. The problem involves conditioning on whether the interchanged ace is selected, which is crucial for determining the correct probability. Participants reference a previous solution found on Physics Forums, indicating that understanding the hint is essential for solving the problem effectively. The conversation emphasizes the importance of articulating the proof using conditional probability notation, specifically P(A|B).

PREREQUISITES
  • Understanding of basic probability concepts, particularly conditional probability.
  • Familiarity with the notation P(A|B) in probability theory.
  • Knowledge of card deck composition and basic combinatorial principles.
  • Ability to follow mathematical proofs and logical reasoning.
NEXT STEPS
  • Study the concept of conditional probability in depth, focusing on real-world applications.
  • Explore combinatorial probability problems involving card games and similar scenarios.
  • Review the solution provided in the referenced Physics Forums thread for additional insights.
  • Practice writing proofs using conditional probability notation to enhance clarity and understanding.
USEFUL FOR

Students studying probability theory, educators teaching mathematical concepts, and anyone interested in enhancing their problem-solving skills in combinatorial scenarios.

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Homework Statement


a deck of cards is shuffled and then divided into two halves of 26 cards each. a card is drawn from one of the halves; it turns out to be an ace. the ace is then placed in the second half-deck. the half is then shuffled, and a card is drawn from it. compute the probability that this drawn card is an ace.
{hint: condition on whether or not the interchanged card is selected.

Homework Equations





The Attempt at a Solution


so at first, i was totally lost. i did a google search, and found another physicsforums thread that went over this question. (https://www.physicsforums.com/showthread.php?t=174152). the solution for the most part made sense to me. however, I'm confused by the original hint, condition on whether or not the interchanged card is selected. what exactly does that mean? and is there another way to solve the problem, other than the solution on the other page?
 
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hi bennyska! :wink:
bennyska said:
is there another way to solve the problem, other than the solution on the other page?

i doubt it :redface:
the solution for the most part made sense to me

hmm … "for the most part" ? :redface: …

i think the best thing is for you to write out the proof (in your own words) as far as you can, and then we'll see where you're getting stuck! :smile:

(use the P(A|B) terminology)
 

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