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I played Texas Holdem with some friends last night. On one hand, the dealer burned a card, drew 3 aces on the flop, then burned a card and drew the 4th ace on the first river card. My question is, what are the odds of happening? It felt like witnessing a miracle.

@doubled answered this to apply to poker which is appreciated. I wonder now what we might expect if we ignored the poker aspect and did include the burn cards from a completely randomized deck:

  1. Non ace
  2. Ace
  3. Ace
  4. Ace
  5. Non ace
  6. Ace

I would think we'd have the following?

48/52 * 4/51 * 3/50 * 2/49 * 47/48 * 1/47
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    $\begingroup$ You will likely get better answers if you could rephrase your question without the poker-specific terminology. $\endgroup$ Jun 15, 2020 at 15:19
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    $\begingroup$ You witness a miracle in every game. For instance, last night I saw a game in which the flop and the first river cards comprised the two of spades, the three of spades, the seven of diamonds, and the Jack of hearts. What were the chances of that?! $\endgroup$
    – whuber
    Jun 15, 2020 at 17:10
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    $\begingroup$ My comments are intended to be instructive, not snarky, so I'm sorry that you are reading them that way. A deeper issue concerns the meaning of "entire scenario:" would that be observing four aces exactly as you initially described? Would it mean observing four aces by the end of the game? Observing four aces during one of the night's games? Observing four of any kind in any of those situations? Arguably, the most pertinent characterization is "what's the chance of observing something during a night of poker that looks unusual?" The answer to that is almost 100%. $\endgroup$
    – whuber
    Jun 15, 2020 at 20:51
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    $\begingroup$ Also, to respond to your comment about @whuber's comment, I think such comments are indeed thought-provoking, and more generally, maybe I'm too optimisitic, but I think one of the virtues of sites like this is one is that typically, people genuinely want to help by providing new ways to approach a problem. When I first learned about probabilities, I think a comment like that one would have really helped me understand what we really mean by 'low probability events.' 4 Aces is awesome, but so is any 4 of a kind, and similarly seeing A-K-Q-J-T suited, or indeed any straight flush, would be cool:) $\endgroup$
    – doubled
    Jun 15, 2020 at 21:51
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    $\begingroup$ @whuber apologies, tone is easily lost in text. Appreciate the input! $\endgroup$
    – Andrew
    Jun 16, 2020 at 17:08

1 Answer 1

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This is the same as the probability of successively drawing any 4 specific cards in a row (as @whuber points out, seeing 4 Aces is just as likely as seeing any 4 specific cards, and what makes Aces special is that we as poker players put weight to that specific realization of cards). We have 4 aces to draw from for the first card, 3 for the second, 2 for the third, and 1 for the fourth. Each time, there is one less card in the deck left, and conditional on adjusting for the number of cards left, each probability is independent, and so the probability is simply: $$P(\text{drawing 4 aces in a row}) = (4/52)*(3/51)*(2/50)*(1/49) \approx .00037\%$$ I multiplied the answer by 100 to get a percentage. Note that this assumes that the deck was truly randomly shuffled, which in friendly play is almost never truly the case (indeed, this may be one reason why we often 'burn' a card, and this randomness assumption is also why the burning of a card has no effect on the probability), but regardless, a crazy and fun event to happen :).

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