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I have been struggling with this question.

Let's say we have $m$ decks. One deck contains all (52) red cards. The rest of the decks contain 26 red and 26 black cards.

Given that I draw a card and it's red, what is the probability it was drawn from the biased deck (containing all red cards)?

My approach: apply Bayes' rule.

$P(d=b| c=r) = \frac{P(c=r|d=b) P(d=b)}{\sum_i(P(c=r|d_i) P(d_i)}$

But, my answer is in terms of $m$. Is that correct?

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  • $\begingroup$ Think for a moment about the qualitative difference between the situation where $m=1$ and $m=1\,000\,000$. Doesn't that convince you the answer must depend on $m$? $\endgroup$
    – whuber
    Commented Oct 21, 2016 at 19:45
  • $\begingroup$ It should depend on $m$. Clearly as $m$ gets larger the probability should tend to zero, for instance. $\endgroup$
    – dsaxton
    Commented Oct 21, 2016 at 19:48
  • $\begingroup$ Are we assuming that the deck is selected randomly (uniformly) from the set of all decks (this seems to be assumed, but should be mentioned). $\endgroup$ Commented Oct 22, 2016 at 6:38

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You are exactly correct! Imagine if you had a million decks, the chance that it came from the biased one (1/million) is tiny compared to the 1/2*(1-1/million) $\approx$ 1/2. The total red cards in the others dominate.

Conversely if you only had 1 deck, it would be 100%!

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