Basic Poker Immediate Odds Question

[Master1022]

TL;DR Summary

Quick question about calculation of immediate odds in a poker hand.

Hi,

Just have a quick conceptual question about calculating immediate odds in poker.

Question: Your hand is Ad Kd. The four cards on the table (flop and turn) are: Kd 7h 4d. What is the probability of getting a flop on the river? (scenario is that we are playing with one opponent)

At first, I thought that this would be calculated by doing $P(diamond\, in\, the\, river) = \frac{\# \, diamonds\, left}{\# \, cards\, left} = \frac{9}{44}$. I have gotten the 9 from the fact that there are currently 4 known diamonds on the table (13 - 4 = 9) and there are 8 cards in play (52 - 8 = 44). However, the blog post quotes a different probability and when I work it back, they seem to use 46 in the denominator. I don't properly have an intuition of why we are including our opponents' cards in the total number of cards?

The answer quoted is: 19.56%

Thanks in advance.

 

[Klystron]

Summary:: Quick question about calculation of immediate odds in a poker hand.

Hi,

Just have a quick conceptual question about calculating immediate odds in poker.

Question: Your hand is Ad Kd. The four cards on the table (flop and turn) are: Kd 7h 4d. What is the probability of getting a flop on the river? (scenario is that we are playing with one opponent)

At first, I thought that this would be calculated by doing $P(diamond\, in\, the\, river) = \frac{\# \, diamonds\, left}{\# \, cards\, left} = \frac{9}{44}$. I have gotten the 9 from the fact that there are currently 4 known diamonds on the table (13 - 4 = 9) and there are 8 cards in play (52 - 8 = 44). However, the blog post quotes a different probability and when I work it back, they seem to use 46 in the denominator. I don't properly have an intuition of why we are including our opponents' cards in the total number of cards?

The answer quoted is: 19.56%

Thanks in advance.

You need to correct your scenario. 1) As Diamond King (Kd) is one of your hole cards, it cannot appear on the flop. 2) You need to show a fourth card on the table after the turn. Also, I presume you mean "probability of completing a flush".

Using 46 remaining cards seems correct. You have seen 2+4=6 cards. 52-6 = 46 remaining cards. You may ignore the "burn" cards since they are unknown. Of those 46 remaining cards, 9 are diamonds.

There are other cards that can improve your hand without making the flush.

 

[FactChecker]

I don't properly have an intuition of why we are including our opponents' cards in the total number of cards?

Because some diamonds might be in the opponent's cards. Since they are unknown, they must be included in the entire set of cards where the remaining diamonds might be.

 

[PeroK]

I don't properly have an intuition of why we are including our opponents' cards in the total number of cards?

The simple answer is because you don't know how many diamonds your opponent holds. Excluding those cards from your calculations effectively assumes your opponent holds no diamonds.

Sometimes to see these things you can take a more extreme example. Suppose you hold 23 cards, with 7 diamonds. And your opponent holds 23 cards. Six cards remain "in the river".

By you calculation the remaining six cards would be all six remaning diamonds! As follows:

$P(diamond\, in\, the\, river) = \frac{\# \, diamonds\, left}{\# \, cards\, left} = \frac{6}{6} = 1$

 

[Master1022]

You need to correct your scenario. 1) As Diamond King (Kd) is one of your hole cards, it cannot appear on the flop. 2) You need to show a fourth card on the table after the turn. Also, I presume you mean "probability of completing a flush".

Using 46 remaining cards seems correct. You have seen 2+4=6 cards. 52-6 = 46 remaining cards. You may ignore the "burn" cards since they are unknown. Of those 46 remaining cards, 9 are diamonds.

There are other cards that can improve your hand without making the flush.

Yes, sorry, I mixed up the terminology - I was looking for completing the flush

 

[Master1022]

OKay, thank you all for posting - that does clear things up!