I am a mediocre poker player and doubt I could become a good one even if I worked hard at it, which I have no interest in doing. I play because it’s fun, and I keep the stakes low enough that I don’t mind that I usually lose. Also, I find poker enjoyable to write about in terms of human psychology, for example how a logical puzzle about a poker game has analogies to difficult romantic relationships and how a memorable bad luck outcome can tempt us to revise decision rules that are in fact generally reliable. I played in a hand not long ago that fascinated me in terms of the power of suggestion.

In a Texas hold ’em game of eight players, the flop was unusual: 9 of diamonds, 10 of spades and Jack of clubs, in that order. It looked like the dealer was laying down a straight on his own, and even though the order of the flop cards is irrelevant, the fact that the cards came in sequence made it even harder for everyone not to be thinking straight straight straight.

One of the players was in heaven. He had taken a risk by staying in for the flop during the first round of betting with only an 8 and Queen of hearts. With a straight to his name now, it would be very hard to lose and he bet with confidence. Two other players kept betting up to the turn card, which came up a 4 of clubs. One player folded at that point despite having a not bad hand (a 10 of diamonds and Ace of hearts), because he assumed based on the flop that for the other two players to be staying in and betting significantly, there was at least one straight in the offing.

The river card was a Queen of spades. The player with the straight groaned inwardly, but then went through the possibilities for a straight the other remaining player (me) might have. If I had a 7 in the hole and had been hoping for an 8 or had a Queen in the hole and has been hoping for an 8, I was out of luck. In the unlikely event that I was holding an 8 and had been hoping for a Queen (unlikely because he had one of the 8s), it would be a split pot: no tragedy.

He said out loud “The only way I can lose is if you were stupid enough to have a king and be hoping since the flop that a queen could come up”. I took it as a compliment that he bet big after saying this, because it would indeed have been foolish to expect to fill an inside straight like that.

I called his big bet and he turned over his straight. I turned over a king, for a higher straight. Care to guess why I stayed in?

As you might have guessed, my other card was also a king. The flop mentally suggested straight very strongly and of course so did the other player’s own hole cards, so it was cognitively hard for him to break out of that mental frame and realize that I might have been just playing a pair of kings. When the flop came up as it did, I knew my kings would beat anyone who had paired one of the flop cards, so I stayed in. I realized that I would lose to a 7 and an 8 or an 8 and a queen in the hole, but that seemed very unlikely (Even though of course it happened), and I obviously didn’t have to worry about someone holding an 8 in the hopes of landing a queen on the turn or river.

The guy I beat on this hand is a better player than I am. I would have done what he did and lost like he did if not worse. It’s very hard for any of us to not go inexorably down a particular line of reasoning when all signals seem to be pointing there so strongly.

Gee, and I was expecting an atrocious pun of some kind!

But since this post has something to do with games of chance, I want to ask someone with some statistical know-how to take a look at some figures from recent Powerball drawings and resolve an anomaly that I cannot figure out.

The data come from two websites: one, http://www.lottoreport.com/ticketcomparison.htm , reports the numbers of tickets sold on each of the draw dates for Powerball, and a second one, http://www.powerball.com/powerball/pb_winner_summ… , reports the numbers of winning tickets not only for the Grand Prize but also for the smaller prizes. You win the jackpot by matching all five white balls in the drawing plus the Powerball, which is drawn separately. But you can win lesser amounts by matching fewer balls, and the odds of these smaller winnings are much greater than for the jackpot where the odds are 292 million to one against each ticket.

The largest of the lesser prizes is "Match 5," involving tickets with all 5 white balls but not the Powerball. The odds against this are 11,688,054:1. The second largest of the smaller prizes is "Match 4 plus PB," with odds of 913,130:1. There are additional prizes of lesser amounts but these two suffice to illustrate a problem I am trying to understand. It appears that there are some large divergences between the observed and expected numbers of winning tickets for several on the prize categories.

On Jan. 9, where 440 million tickets were sold, there were 28 "Match 5" tickets. The expected value for this was 37.67, and the cumulative binomial distribution was 0.0625, which presents no problem. That is well within the probability predicted under that distribution. Almost all of the values for Match 5 in recent months have been between the statistically expected 0.025 and 0.975. On Jan. 13, though, with 635 million ticket sales, there were 81 Match 5 tickets, with an expected value of 54.34, and the cumulative binomial for that was 0.99972.

That seems a bit anomalous but "Match 4 plus PB" seems even more so. For most drawings up to and including Jan. 9, the cumulative binomial has been "well-behaved." On Jan. 9, the expected number was 482 and the observed number was 451, with a cumulative binomial of 0.08. No problem there. But on Jan 13, the expected number of Match 4 plus PB was 695.5 and the observed value was 934 (107 of which elected the Power Play to double their monetary return). For this value of 934, Excel returns a cumulative binomial of 1.0, which appears to be impossible, but it is, after all, 9 standard deviations from the mean of the binomial. Even using the value of 827 non-Power Play tickets, which is 5 SD from the mean, is far too unlikely for this category.

It appears at first glance that there were "too many" winners of lesser prizes on the latest drawing. Were there? For several other categories of smaller prize winnings, using the same databases of tickets sold and numbers of smaller prize winnings, there appear to be rather large differences between the observed and expected numbers of reported winning tickets.

I am probably overlooking something obvious here, but I am at a loss to find it. What am I missing? The data for the numbers of prizes comes from the website of the multi-state lottery commission, and should be accurate; the number of ticket sales seems to be similarly reliable. The mean of the binomial distribution is the number of tickets times the probability of a particular prize category. The variance of the mean is “npq,” and the standard deviation is the square root of the variance. The difference between observed and expected winning tickets is the numerator of the Z statistic, the standard deviation is its denominator, and the larger the absolute value of Z, the more improbable is the result. It all seems so sound, but for numerous drawings in the past few months, the numbers of winning tickets for smaller payoffs are quite different from what the probabilities would predict.

I am grateful for any light anyone can shed on this vexatious problem. If valuable for nothing else, the principle I am overlooking could perhaps serve as an interesting classroom exercise for introductory courses in biostats.

A contributing factor may be that, though the numbers drawn are random, the numbers that players pick are often aren't.

Keith, your opponent might not be as good as you/he thinks he is. You wrote: "One of the players was in heaven. He had taken a risk by staying in for the flop during the first round of betting with only an 8 and Queen of hearts. With a straight to his name now, it would be very hard to lose and he bet with confidence."

Hmmm … KQ is a good hand pre-flop. Anybody holding that would likely stick around in the face of a three-bet, so long as there were several players in the pot. So his Q8 is not so hard to lose after all, is it?

Well of course he did lose this particular hand so you are right, although most of the time he would have won so I think it is still right to say that it was hard to lose.

As to whether I over-estimate his abilities, it could be. I am a bad player, so all I can really know is that he’s not as bad as I am.

I presume, with KK you raised and reraised. Then the Q8 suited should have folded. Poor play on his part. In the long run that kind of play is a certain loser.

Better players would be wondering what you had before the flop and get that in their mind.

Actually I slow played it because the other players were raising and I wanted to keep them in rather than scare them off. We were playing table stakes and I assumed in the end it would all be in the pot and it just about was.

This (rather than the power of suggestion) is likely why he was surprised by your hand. It would have been hard to put you on KK or AK if you didn't raise, re-raise preflop.

Oh dear, slow playing KK. That is alright with couple of players in, but in a 8 handed post flop you are the favourite only about 35% of the time. But your Q8 guy really made some other mistakes besides staying in. After the river Q, and moaning about it a big bet was pointless on his part. No one would stay in unless they had him beat. So he is just throwing money away. He sounds like a guy who plays his cards without paying attention to what anyone else may have. Common but a good way to lose.

Anyways this has been an entertaining post!

Thanks for this you clearly have a better grasp of the game than I do — there is no false modesty in my saying I am a bad at it (Although your "no one would stay in" isn't quite true — don't spread this around but I have been known to steal big pots will ridiculous bluffs now and then). I didn't describe the details fully in the post but I was late in the pre-flop order, with one raise, one call and 2 or 3 other people having folded before it came to me. So I was thinking someone probably had something but my own something was likely to be better so I just called. That may have been a mis-estimation of the odds and perhaps I should have raised myself to force others out.

But in any event, I am glad you enjoyed the post and I appreciate your contributions to my education in the game.

There's always more to learn, which is why it's fun to play.

"So I was thinking someone probably had something but my own something was likely to be better so I just called."

You were almost certainly right (seeing as how only AA is better), but from that it follows that you should raise. If they call, then you are getting their money while you are ahead. If they fold, it improves your odds of winning the pot. Either way, you come out ahead. Otoh, calling allows folks to see the flop cheaply and potentially catch up (which is what happened).

Thanks again for the post!