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This is a question that came up with my friends while playing a card drinking game:

You draw cards without replacement from a standard 52-card shuffled deck. What is the probability that you draw all of the kings before drawing any of the jacks?

I was thinking of a solution along the lines of combinations, but I'm not sure if that's the way to go. Simulation tells me the answer is ~1.4%, but I don't know how to get to this answer.

Disclaimer: This is not a homework question. I'm really just curious.

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  • $\begingroup$ You should never drink while playing cards. $\endgroup$
    – wolfies
    Commented Apr 11, 2016 at 14:23

1 Answer 1

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The only thing that matters is the relative placement of the two critical types of card.

There are $\binom{8}{4}$ equally likely ways to choose the places occupied by the Kings. Only one of these is "favourable."

Thus the probability that all the Kings come before any Jack is $\dfrac{1}{\binom{8}{4}}$. This is approximately $0.0143$, quite close to the number obtained in the simulation.

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  • $\begingroup$ Of course, that's so simple. Thanks! $\endgroup$
    – Marvin
    Commented Apr 11, 2016 at 5:20
  • $\begingroup$ You are welcome. $\endgroup$
    – André Nicolas
    Commented Apr 11, 2016 at 5:20

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