What Are the Odds of Hitting A, J, or 2 on the Flop in Poker?

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Discussion Overview

The discussion revolves around calculating the probability of drawing an "A", "J", or "2" on the three-card flop in poker. Participants explore different assumptions regarding the cards dealt to players and how these affect the probabilities. The scope includes mathematical reasoning and probability calculations relevant to poker scenarios.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant calculates the probability of drawing "A", "J", or "2" on the flop as 0.705 by summing individual probabilities for each card drawn sequentially.
  • Another participant agrees with the initial probability for the first card but argues that the calculations for subsequent cards should consider the cards already dealt, suggesting a different approach to the probabilities.
  • A third participant reiterates the need to adjust probabilities based on the known cards dealt to players, proposing that the probability for the first card on the flop should be calculated as 12/50 if no "A", "J", or "2" has been dealt.
  • One participant questions the assumptions regarding the knowledge of the dealt cards and asks for the probability of at least one of the three flop cards being "A", "J", or "2" without knowing the players' cards.

Areas of Agreement / Disagreement

Participants generally agree that the probability does not depend on the number of players but disagree on how to calculate the probabilities based on the cards dealt. The discussion remains unresolved regarding the correct method for calculating the probabilities.

Contextual Notes

There are limitations regarding assumptions about the cards dealt to players, which affect the calculations. The discussion highlights the complexity of calculating probabilities in poker scenarios, especially when considering known versus unknown cards.

Offramp
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What is the probability of anyone one of "A, J, 2" showing up on the the 3 card flop. I am assuming the number of player is 10 or less. I am assuming the probability is the same if it's 1 or 10 players , right? This is what I've calculated. Please let me know if I am correct.

Prob of 1st card being A, J or 2 is 12/52 = 0.230
if 1st card is none of the above then,

Prob of 2nd card being A, J or 2 is 12/51 =0.235
if 2nd card is none of the about then..

Prob of 3rd card being A, J, 2 is 12/50= 0.240

I would then sum up the prob. which is 0.705 am I correct?
 
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While the probability does NOT depend on how many players there are, it does depend on what information you have: in other words what cards you have been dealt before the flop.

Assuming that you were not dealt any "A", "J", or "2", then, yes, the probability the first card is "A", "J", or "2" is 12/52= 0.230 and the probability that it is NOT is 48/52.

The probability that the first card is NOT one of those and the second card IS would be 12/51 times 48/52= 0.217 (approx.). The probability, in this case, that the second card is NOT one of those would be 1-0.217= 0.783.

The probability that the first two cards are NOT one of those three but the third card is would be 12/50 times 0.783= 0.188.

The probability that exactly one of the three cards is one of "A", "J", "2" is 0.230+ 0.217+ 0.188= 0.635.

The probability that at least one of the three cards is "A", "J", "2" would be much more complicated to calculate.
 


HallsofIvy said:
Assuming that you were not dealt any "A", "J", or "2", then, yes, the probability the first card is "A", "J", or "2" is 12/52= 0.230 and the probability that it is NOT is 48/52.
I don't think that is quite right: since you've been dealt two cards, you know what two of those cards are. If you've not got one of A J 2, then the probability of A J or 2 being the first card on the flop is 12/50. (etc.. for the other cards of the flop.)
 


assuming that the two cards dealt to the players are NOT KNOWN, then what is the probability that atleast one of the 3 cards on the flop is a A, J, 2?
 

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