Video Poker Forum

I attempted to answer this with the following calculation. However, only getting the third card once in 7.66 times seems pretty low. Is this correct?

(Remember I am dealt a pair, I want to know the probability of getting 3 of a kind on the draw.)

2/47 + 2/46 + 2/45 = 1/7.66

Only 3 of a kind? Or 3 of a kind OR 4 of a kind?

If it's only 3 of a kind, then you have to also look at NOT getting the 4th one in the other two slots, and the 3rd could be in any of the remaining 3 slots. So I think it's more like:

(2/47 * 45/46 * 44/45) + (45/47 * 2/46 * 44/45) + (45/47 * 44/46 * 2/45)

That is, the odds of a matching card times the odds of a non-matching card in the other two slots, for each slot.

Which actually ends up being the same as ((2*44*45) / (45*46*47)) * 3 = (3960 / 97,290) * 3 = .1221 or roughly 12.2%

Not certain though.

Almost, Eduardo, but you're leaving in the full house outcomes. You have not only to eliminate the card that would make a quad, but also any card that would match the first non-matching card. Unfortunately, the odds of that are different depending on whether the first non-matching card had the same rank as one of your discards.

The easiest way to answer questions like these is to use Video Poker for Winners or the Pro Option at this website. You could then see quickly that the number of trips drawing to a pair is 1854 out of 16,215.

If you want to match it up with your notation, we can note that the remaining cards in the deck are 9 cards matching the ranks of our discards, 36 cards of fresh ranks, and 2 cards matching our pair. So, above you need to separate the 45 non-matching first draw into 9 and 36 and then use either 41 or 42 for the number of non-full house, non-quad draws for your 2nd non-matching draw. When your first non-matching draw was a rank that you had discarded, there would be 36 + 6 = 42 safe draws; when your first non-matching draw was not a rank you had discarded, there would be 32 + 9 = 41 safe draws. "Safe" here means leaving the ability to have trips as the final outcome.

Going through the same steps that Eduardo did above armed with the knowledge of eliminating full houses and dealing with the extra category of ranks, we get

3*(2*9*42 + 2*36*41) / (45*46*47) = (9*42 + 36*41) / (15*23*47) = 1854/16215 = 0.1143

For simplicity sake I can say 3 of a kind OR 4 of a kind.

Basically I was getting frustrated getting dealt many low pairs that never amounted to even a 3 of a kind. But if I'm only supposed to get 3 of a kind or better 1/7.66 attempts (that's less than completing a flush or inside straight draw!) I guess I should not be disappointed.

Holding a pair and discarding 3 singletons allow the following improvements:

2592 Two Pair
1854 Three of a Kind
165 Full House
45 Four of a Kind

The remaining 11,559 possibilities are no improvement over your pair.

If you want probabilities, divide by COMBIN(47,3) = 16,215

3 of a kind? I only care about getting 4 of a kind with a kicker though

Thanks for the responses!

However I am curious now, what are the odds of drawing the quad in this scenario?

(Btw- nice screenshot hophoofer! I never heard of "triple *Triple*" but it looks like you get the same as a royal for your hand at the expenses of straights only being worth 3...right? Regardless congrats!)

Oh thank you!

The odds of completing a quad holding a pair is

(47 - 2) / COMBIN(47, 3) = 1 in 360.33.

9/5 TTB pays 98.6%, 9/6 TTB pays 99.8%. Everyone should play 9/6 TTB if you can find one.

Thank you for the calculation. Is there a good guide for doing these calculations on our own? (I took a probability class in college, but I don't remember enough to do too much.)

I will look for TTB - the volutility must be through the roof though! Still I'll check it out.

hophoofer wrote: ↑

Wed Oct 28, 2020 6:15 pm

Oh thank you!

The odds of completing a quad holding a pair is

(47 - 2) / COMBIN(47, 3) = 1 in 360.33.

9/5 TTB pays 98.6%, 9/6 TTB pays 99.8%. Everyone should play 9/6 TTB if you can find one.

Correct --- however in this game it matters if the fifth card is an ace, a 2 or 4, or one of the other cards. It's not a tough calculation (software makes it easier), but it matters what you throw away. That is, if you drew from 33KQ8, your odds are better for a special quad than if you drew from 33A24. Just looking at the final hand, you can't tell.