Ask The Wizard #246

What is the effect of player mistakes in blackjack?


Peter Griffin devoted a whole chapter to that question in his book Extra Stuff — Gambling Ramblings. His study was based on observing 11,000 actual hands of play in 1987. The following table summarizes his results of the cost of errors made.

Cost of Blackjack Errors

Location Cost of Errors Margin of Error
Atlantic City 1.13% 0.12%
Las Vegas 1.67% 0.17%
Reno 1.48% 0.19%
Lake Tahoe 1.39% 0.54%
Total 1.41% 0.10%

In my opinion, play has improved a lot in the 23 years since the study. If forced to guess, I think the cost due to errors is about 0.5% now. I would agree with Griffin that Atlantic City players are more skilled than Vegas players.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

After this column first appeared, I heard from gaming consultant Bill Zender. He offered to let me post his article How Poor are Blackjack Players (PDF 147K), from Gaming Operations magazine. There is says his own research showed the cost of player mistakes to be about 0.83%.

I really like the nice compact guides at your Wizard of Vegas site. I also have a question about the sports betting guide. The board pictured has no total or money line for a couple of the games, so I assume the casino was not offering those bets. Is that related to the size of point spread or is there some other reason they are not there?


That picture was taken at a Leroy’s franchise, which is a small and conservative sports book. They tend to put out the spread first, and later the total and money lines.

It is also unusual to offer money lines on significantly lopsided games, for example the Eastern Michigan vs. Central Michigan with a 23-point spread. It is difficult to set accurate money lines with such games, and the risk of unbalanced action is greater. The best casino for betting money lines on games with big spreads is the Hilton. Finally, if the point spread is very small, like 1 or 1.5, then most sports books don’t bother to post a money line at all, because betting against the spread would likely result in the same outcome.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

My question is based on dice odds. I know that there are six ways to get 7 and one way to get 12, but what are the chances of getting six 7’s before one 12? Are they even, and if not, how many twelves should be added to the equation to make it an even proposition?


The probability of rolling a 7 is 1/6, and the probability of rolling a 12 is 1/36. The probability of rolling a 7, given that a roll is a 7 or 12 is (1/6)/((1/6)+(1/36)) = 6/7. So the probability that the first six times a 6 or 12 is rolled it is a 6 every time is (6/7)6 = 39.66%.

If you rephrase the question to be what is the probability of rolling five 6’s before a 12, then the answer is (6/7)5 = 46.27%. With four rolls it is (6/7)4 = 53.98%. So there is no number of 7’s before a 12 that is exactly 50/50. If you’re looking for a good sucker bet, suggest you can either roll four 7’s before a 12, or a 12 before five 7’s.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

Hi, I understand you’re busy so no worries if you can’t find the time to respond. I’m currently a math major at UNLV and am looking for some kind of direction relating to what to do with my life once I graduate. I’ve searched quite a bit for this information, and have definitely come up short. It seems the general idea here is that a math bachelor’s degree is quite useless, and that I should have been an engineer instead of a mathematician. I’m not very interested in being an actuary, because I don’t want to take additional time to pass the exams. Since I will probably stay in Vegas after I graduate, it seems that a great idea would be to work for the gambling industry, which you seem to have much experience with. Could you offer me any advice or direction?

Christina from

I can sympathize with your dilemma. When I graduated in 1988 with a bachelor’s degree in math and economics, I had no idea what to do with it. The actuary exams looked daunting at the time, because I didn’t study statistics deeply at UCSB, and somehow I didn’t feel actuarial work was my calling in life. However, after accomplishing absolutely nothing for a solid year after graduation, I gave a second look at the actuary exams. With the help of a tutor, I gave the exams the old college try, and passed the first three with high grades in a little more than a year. That eventually led to nine years of actuarial work at the Social Security Administration, where I finished the rest of the associate exams. That said, I offer the following advice.

First, don’t dismiss the actuarial field lightly. If you are smart enough to get a math degree, you are smart enough to pass the exams. Maybe you’ll need to teach yourself some statistics, but I think you can do it. Actuarial work may not seem terribly exciting, but the profession is consistently highly rated for good working conditions and compensation.

Second, at least consider the National Security Agency. They hire lots of mathematicians to analyze a sea of chatter gathered from listening in on phone calls all over the world. If you take their screening test, don’t volunteer too much information or let them fluster you. This comes indirectly from people who flunked it.

Third, most mathematicians in the gaming business work for the big slot machine companies. There are not a lot of jobs in that field, but not that many people know such jobs exist either. Contrary to what many people think, the casinos don’t hire designated mathematicians. Overall, I think actuarial work is better paying and more interesting.

Finally, don’t put down a math degree. You’ll probably never use what you learned in the more advanced math classes, but the accomplishment itself shows you should be able to learn what you’ll need to for an entry-level position in any one of the fields I mentioned.