Top Two Pair Odds Example

You get dealt a King of Diamonds and a Nine of Hearts. The flop is lookin' pretty good...

Top two pair!

Lesson 1: What are my chances of getting a full house on the turn?
To get a full house, you'll need another King or Nine to pop up. There are presumably two of each left in the deck. So you've got 4 outs. After the flop, there are always 47 cards unaccounted for. 4/47 is around .085 or an 8.5% chance of you getting that boat.

Lesson 2: What are my chances of getting a full house on the river?
If it didn't happen on the turn, your chances usually don't change all too much, but let's check. You've still got 4 outs and now 46 unseen cards left. 4/46 is about .087 or around an 8.7% chance of hitting it on the river. A .2% difference. Sorry.

Lesson 3: How about the chances of getting the boat on the turn OR the river?
Like the previous examples, to figure your chance of something happening on multiple events, you need to calculate the chance of it NOT happening first. On the turn, it won't happen 43/47 times. On the river, it won't happen 42/46 times. 43/47 is .915, and 42/46 is .913. Multiply them and get .835, or 83.5% chance of it not happening. Subtract that from 1 and you get a 16.5% chance of getting at least a full house by the showdown.

Lesson 4: What do you mean by "at least"?
Since we figured the chances to NOT get dealt a full house, the chances are built in if the turn and river are two Kings, two Nines, or a King and a Nine. If you are dealt two cards both of either King or Nine, it'll be four-of-a-kind and not a King and Nine 33% of the time. Think of it as being dealt one card then the other. What are the chances of the first card matching the second? Whether it's a King or Nine, there will be only one unaccounted for, but two of the other. That's 1/3, or 33%.

Lesson 5: Then what are my chances of getting four-of-a-kind?
This is a little more abstract. I hope I warmed you up for this with the previous lesson.
It doesn't matter which card we're banking on. We need to first get a full house on the turn. According to lesson #1, the chance of that happening is .085. The chance of getting the same card we got on the turn is 1/46. There's only one out, and the usual 46 unseen cards. 1/46 is around .022, or 2.2%. Multiply the two probabilities (.022 X .085) and get .002 or one-fifth of a percent. It will be Kings half of the time and Nines the other half.

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