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BGonline.org Forums
Solution
Posted By: Timothy Chow In Response To: Tricky probability puzzle (Timothy Chow)
Date: Monday, 30 October 2023, at 12:46 a.m.
The key question is whether a non-ace is just as likely to be dealt as an ace. What's the probability of being dealt the ace of spades, for example? If the first ace is the ace of spades, then certainly you're not going to be dealt the ace of spades. If we focus only on this fact, then it might seem that the ace of spades is less likely to be dealt than (say) the two of clubs. But in the event that the first ace is not the ace of spades, then the chances of being dealt the ace of spades are elevated (compared to the two of clubs), because the ace of spades has to come after the first ace, whereas the two of clubs might have already been skipped. Do these two effects cancel each other out exactly?
Yes. Here's how Mind Your Decisions explained it, which I think is a nice argument. Imagine that the way the deck was initially shuffled was to set aside the ace of spades, and shuffle the remaining 51 cards. Then as a last step, the ace of spades was randomly inserted into one of the 52 possible positions in the deck. Though this procedure might seem a little strange, it does result in all possible orderings of the cards being equally likely, and so is a valid way to shuffle the cards. The crucial observation is this: of the 52 possible ways to insert the final card, exactly 1 way results in its being the card immediately after the first ace. Therefore the probability that the card is dealt is 1/52. But this argument holds regardless of whether the card you set aside initially was an ace or a non-ace.
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