This week's ESPN scoreboard

I just finished watching this week's WSOP coverage on ESPN. Once again, they fed us two "poker facts" which need checking. (For the earlier ones in this series of posts, start here and work backwards following the links, or just type "ESPN" in the search box at the top of this page.)

Here's the first alleged fact:




Let's assume that their inelegant wording means three of a kind on the flop, rather than after the flop, for the sake of clarity. I'm also going to cut them a little slack and assume that "full house or better" actually means "full house or four of a kind." That is, I'll assume that they did not intend to also include straight flushes, the probability of which would vary a lot depending on the exact cards in question. They just didn't phrase their query very precisely.

Once again, this is easier to check by figuring out the probability of not making a full house or quads on the turn or river. Suppose we start with A-K and see a flop of A-A-Q. That's five cards gone from the deck, leaving 47. Seven of those will make a full house or quads on the turn (one ace, three kings, and three queens), so the probability of not doing so is 40/47. If we don't make it on the turn, then 10 out of 46 remaining cards could make such a hand on the river (the same seven as before, plus the three pairs to the turn card), so the probability of not making it is 36/46. Multiply those two fractions together and we get 0.6660, or 66.60% as the probability of not making a full house or quads after having flopped three of a kind. Therefore, the probability of making it is 33.40%.

Hey, that's what ESPN said! Amazing!


OK, maybe they can get both right this week. Here's "fact" #2:




So far in this series, all of their facts have been specifically about hold'em. Sadly, ESPN is wrong here, as far as hold'em is concerned.

The number of possible ways to pull seven cards from a deck of 52 is given by C(52, 7), which Excel will instantly calculate for you as 133,784,560. We don't care what order they come in. Let's look at just the probability of getting a specific royal, such as in clubs. Of the seven cards, five are specified to be the royal, leaving two slots that can be filled by any other cards. There are 47 cards left that could fill one of those slots, then 46 left for the other. That makes a total of 47 x 46 = 2162 seven-card hands that contain a club royal flush. Now multiply that by four so that we encompass all four suits, and there are 2162 x 4 = 8648 seven-card hands that contain a royal. With 8648 hands containing a royal, there are 133,784,560 - 8648 = 133,775,912 that do not contain a royal, so our odds are 133,775,912:8648, or 15,469:1. Those are the odds against hitting a royal flush on any hold'em hand if you see all seven cards--actually a lot better than ESPN is reporting.

So where is ESPN getting its number? It appears that they are working with a five-card poker hand, such as five-card draw. Alternatively, they could have intended to figure the odds of flopping a royal flush, or seeing one come on the board of five community cards. If that is the question, then they have it almost, though not quite, right:

The number of five-card poker hands is given by C(52, 5), which is 2,598,960. Of those, obviously only four are royal flushes, and 2,598,956 are not. That makes our odds 2,598,956:4, which is 649,739:1. ESPN said it was 649,740:1. This is not a rounding error on their part. Instead, they probably forgot to subtract the four royal hands from all possible hands before taking the ratio, so they figured it erroneously as 2,598,960:4, which would be 649,740:1. Close, but still an error.

More importantly, though, since every previous poker "fact" has been about hold'em, and they are adding this little feature into a broadcast hold'em event, I think the viewer has a reasonable expectation that the "facts" presented are about hold'em, when the answers would differ among the various forms of poker. That means that ESPN needed to do the math with a seven-card hand, rather than a five-card hand. (That's slightly sloppy language on my part. I realize that even when one has seven cards to choose from, one's final "hand" is made up of only five. There isn't really anything such as a "seven-card poker hand." But I don't know any other reasonably compact phrase for describing what I'm meaning, so please forgive the slight imprecision.) For example, the first "poker fact" discussed above has to be hold'em; the answer would be different for Omaha, even though both use a flop, turn, and river.

Since ESPN erred both in using a five-card hand rather than seven (or not specifying a different game), and got the math wrong even assuming a game such as five-card draw, I have to score them a zero for this one.

ESPN's score for this week: One hit, one miss.