Poker: starting pair and flopping quads probability?

Question

I've been looking at tables with the probabilities of flopping certain hands and I took to my calculator to try and see if I could figure them out myself.

I'm having trouble with the flopping quads with a starting pair. According to these tables, the probability of having a starting pair and flopping quads is 0.25% or 400 to 1. I don't seem to get this answer at all.

So if you already have a pair, then there are 2 outs with 50 remaining cards in the deck. So I did:

1-((48/50)x(48/49)x(47/48)) (assuming we hit an out on the first card of the flop, but I guess it doesn't really matter whether we get the 1st out on the first or second card, right?).

This gives me 7.9% or 11.8 to 1, which is way off what they have given.

So I'm assuming they factor in the probability of even getting dealt a pair to start with, which is 5.8% or 16.24 to 1 [(13C1)x(4C2) / 52C2].

How do you work it out?

Answer

You're off to a good start -  you've got a pair of Aces. 
 Now to make quads you need the other pair.  Once that happens you don't care what the last card is, you're too busy ordering drinks for the house.    

The math looks roughly like this.  ((2*1)/(2*1))*48 = 48    

Your pocket pair accounts for two of the the four suits, let's say hearts and spades.  The means your flop must contain both the Ace of clubs and the Ace of diamonds.  The first part of the equation accounts the choosing from the two remaining aces.  No matter where they appear in the flop, there is only ONE combination of the two suits.  Diamond - Club being the same combination as Club - Diamond. That one combination of the final two aces can then be matched with any of the 48 remaining cards.  Thus you have 48 flop combinations that will complete your quads out of the total of (50C3) or (50*49*48)/(3*2*1) or 19,600 flops.  That gives you odds of (19600/48)-1 or 407.33 to 1   

Hope the helps.  I'll have a beer.

Poker: starting pair and flopping quads probability?

Answers (1)