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Q: You have a 5 decks of cards numbered 1-12. You draw 4 cards at random from each deck. What is the probability that the 4 cards come out in an increasing sequence from at least one of the draws?

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A: Note a subtle point here. The number of cards in the deck to draw from doesn't really matter. All we are concerned with is if the numbers are in an increasing sequence. If you draw 4 cards from any sized deck, you are left with 4 distinct cards and that's it! Given 4 cards there are \(4! = 24\) ways (permutations) to place them in any given order. Thus, the required scenario happens with a probability of \(\frac{1}{4!} = \frac{1}{24}\).

Therefore,

- The probability that they are
__not__in an increasing sequence is \(1-\frac{1}{24}= \frac{23}{24}\). - The probability that all 5 draws from each of the 5 decks is
__not__in an increasing sequence is \(\big(\frac{23}{24})^{5}\) - The probability that at least 1 of the 5 draws from each of the 5 decks is in an increasing sequence \(= 1 - \big(\frac{23}{24})^{5}) = 19.16\%\) which is the sought probability.

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