I've always been good at math. Well, at least for as long as I can remember. When I was 8 or 10 years old, this meant that I could quickly add numbers together. When I got to my pre-teens, it meant that I could quickly multiply numbers together. I still have these skills. I pretty much take them for granted and admittedly sometimes am amazed that most everyone else can't do the same. I don't expect everyone to be able to multiply two 2-digit numbers together, but the numbers of times I'll hear someone ask something like what's 500 divided by 25 and not be able to do the math in a split-second is mind boggling to me.
I'm not trying to be snooty about things. I'm sure this is the case for anyone who is good at anything. It comes so naturally that you just can't figure out why most people can't do it. Given what I do for a living, I'm probably better off that people aren't so adept at math. It leaves me with less competition! I've always been willing to take the time to try and explain any concept to anyone who wants to listen and/or asks the question. This past week, I dealt with something like this twice.
The first person had a question about Royal Flushes at a full-table of Texas Hold'em. Was the frequency altered because the Player's used 5 community cards. In short, the answer is no. The long-term frequency is not changed, but the volatility is. While the odds of two Players getting a Royal Flush in a 7-card deal (without community cards) is quite rare, it goes to absolutely impossible in a Texas Hold'em game. The only exception is if the community cards form a Royal, in which case all NINE players will have a Royal, not just two. 1 out of 21 Royals at a Texas Hold'em table will form this way. At that very moment, nine Royals will happen simultaneously. How does this not affect the overall frequency? Well, because of all the other times. As I just said, there is zero possibility that only two Players get one in the same hand and unless the community cards contain at least three to the Royal, there is ZERO chance. In essence the Royals will clump together a bit more than with no community cards (when the community cards form a Royal) and they will be a bit more rare the rest of the time. The long-term result is that the frequency of a Royal will be the same, they'll just group together at times.
The person who posed this question to me had some idea of the answer, but wasn't completely sure so he ran the question by two math guys (myself and someone else). We gave him the same answer. This is one of the less frustrating situations where someone is willing to ask the question.
Unfortunately, the other situation was a bit more exasperating. I received an e-mail expressing concern that a particular game with two independent sidebets had a problem. Each of the sidebets on their own was fine, but if the Player wagered on both of them, then he transformed the game so that there was a Player advantage. My very first reaction was NOT A CHANCE. First of all, if this was the case, they wouldn't be independent sidebets by their very definition. To make matters worse, the e-mail contained an attachment both explaining the issue in words and a spreadsheet walking through the math. A casino was frantic and we needed to confirm that there, in fact, was no Player advantage.
This was one of those cases where a little bit of knowledge is a dangerous thing. This was not someone merely inquiring about the possibility of something. Instead it was a rather someone who hit the fire alarm button and made everyone panic for a few minutes until cooler heads prevailed.
Somehow the author of this faulty analysis came to his conclusion because sometimes the Player wins both sidebets with the same hand. Further, complicating the matter, he decided that wagering 1 unit on each was the same as wagering 2 units on a 'combined' paytable of sorts. I'm guessing that if I showed the spreadsheet that was attached to most people, they'd look at it and say it makes perfect sense and there must be a Player advantage here. Fortunately, this issue finally made it across my desk where I was able to untangle the web that got created.
Lost in the analysis was the situations where the Player won ONE of the sidebets and lost the other. In these cases the Player would win the payout indicated and have the one unit wagered on that sidebet returned (payouts were TO 1). But, he would LOSE the other wager. By combining the paytables and the wager, the math was now altered so that if he won EITHER of the two wagers, the entire 2 units wagered were returned to the Player. This led to a significant increase in the payouts to the Player and led the person to conclude there was a Player advantage.
I would've preferred to deal with this second situation as a simple inquiry rather than a casino in full panic mode. Nothing scares a casino like the potential of having a game with a Player advantage. This is one of those cases where it would've been better to leave the math to a professional.