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Ask the Wizard #213

I was in Palace Station the other day and saw they had a few 7 to 5 double deck tables now. They may just be testing the idea out, but it could catch on, just like 6 to 5 did. So, you might want to include 7 to 5 in your rules variations.


Thanks, that is duly noted. Normally, the house edge under Palace Station rules (double deck, dealer hits soft 17, double after split, re-split aces) is 0.40%. This rule bumps that up to 0.86%. The following table shows the increase in house advantage due to this rule.

7 to 5 BJ Additional House Edge

Decks Increase
1 0.46%
2 0.46%
4 0.45%
6 0.45%
8 0.45%

How many ways are there to roll n six-sided, non-distinct dice? As stated, the dice are non-distinct, so with five dice, for example, 1-1-3-5-6 and 1-6-5-1-3 would be considered the same roll. With two dice, it’s trivial to determine that the answer is 21, but I can’t figure out an elegant, generalized solution.

Don from New York

The answer can be expressed as combin(n+5,n) = (n+5)!/(120×n!). Here is the answer for 1 to 20 dice.

Non-Distinct Dice Combinations

Dice Combinations
1 6
2 21
3 56
4 126
5 252
6 462
7 792
8 1287
9 2002
10 3003
11 4368
12 6188
13 8568
14 11628
15 15504
16 20349
17 26334
18 33649
19 42504
20 53130
21 65780
22 80730
23 98280

Credit to Alan Tucker, author of Applied Combinatorics.

What happens to you if you get backed off/barred from a cruise casino?

Joshua Gavina from South Williamson, KY

That has actually happened to me. If you were backed off, then you just can’t play in the casino any longer. If you were barred, you can’t even enter. I suppose if you did not comply, and were caught, they might make you get off at the next port, and not let you get back on. Signage on my last cruise indicated they will do that if they catch you with illegal drugs.

Playing craps at Barona casino I had three come bets working with odds. After the pass line point won, the come out was a seven so I expected to lose my come bets and have my odds returned to me. Instead, the boxman said for a price I could keep my come bets & odds up. Surprised, I said yes and paid up $15 since the table was hot. Opinion? Is this available in Vegas?

Rick from Poway

I assume the come bets were $5 each. Under this assumption, you were actually making put bets, which you can do at any time, at any casino that allows them. Put bets are generally allowed in Nevada, and generally not allowed in Atlantic City. I normally don’t recommend put bets, because you skip the come out roll on the come bet, which has a 22.2% chance of winning, and only a 11.11% chance of losing. My advice is to start over with new come bets.

I read that Warren Buffet (the world’s third richest man) complained that he only paid a 17.7% federal tax rate, while his secretary paid 30%. This seems outrageous to me. Can you comment?

Joe from Nashville

Normally I would say this is out of my area. However, as a former government actuary for eight years, I know a thing or two about taxes. From what I’ve read, most of Warren Buffet’s income is defined as capital gains, which is taxed at only a 15% rate. Like it or not, the tax laws allow it. What puzzled me is why his secretary was paying as much as 30%. According to this video, he was counting “payroll and income taxes.” By “payroll taxes” he obviously meant Social Security and Medicare taxes. Let’s see if 30% is a reasonable total federal tax rate for his secretary.

In 2007 the highest tax bracket was taxed at 35%, but that only applies to income above $349,700. The income up to that point is taxed much less. Let’s assume his secretary is single, with no dependent children, and her salary was $100,000. First, let’s subtract the minimum deductions. In 2007 the standard deduction for single filers was $5,350. The personal deduction was $3,400. So, we’re left with $100,000 - $5,350 - $3,400 = $91,250 in income subject to income taxes. For single filers in 2007, the tax rate was 10% on the first $7825 in income, then 15% up to $31,850, then 25% up to $77,100, and 28% up to $160,850. So, her income tax would have been =0.1×$7,825+0.15×($31,850-$7825)+0.25×($77,100-$31,850)+0.28×($91,250-$77,100) = $19,660.75. That is only 19.7% of her income. All my assumptions like her income, filing status, and not itemizing worked against her, or for a higher tax rate.

Now let’s do Social Security and Medicare. In 2007, the Social Security tax was 6.2%, up to incomes of $97,500, when it completely shuts off. The 2007 Medicare tax rate was 1.45%, with no cap. So, her combined Social Security and Medicare tax would have been 6.2%*97,500 + 1.45%*100000 = $7,495. Counting those taxes, her overall tax rate would have been ($19,660.75 + $7,495)/$100,000 = 27.2%. Still we’re 2.8% short of 30%.

My best guess is that she is also considering the fact that ultimately she is the one paying the employer’s matching Social Security and Medicare tax. For those who don’t know, Social Security and Medicare taxes are really double that deducted from your checks. The employer pays the other half. However, some, including me, would argue that ultimately it is the employee who pays both. If the employer didn’t have to pay that tax, he would have more money to pay his employees. It is easy to feel that way when you’re self-employed, like I am, and have to pay both shares. If you double the Social Security/Medicate tax, the rate is now ($19,660.75 + 2×$7,495)/$100,000 = 34.7%. I assume the 4.7% difference is because she makes less than $100,000, is married, has dependents, itemizes deductions, or some combination.

The Social Security and Medicare taxes would not apply much to Warren Buffet. First, the Social Security cap of $97,500 would be insignificant to him. Second, those taxes apply to wages, not capital gains, as he defines most of his income to be.

So, that is my best guess as to the math behind Mr. Buffet’s statement.

Update: Shortly after this column appeared I received the following response. In the interests of fairness, I present the following argument that Mr. Buffet is paying too much in taxes.

I read with interest your answer to the ’outraged’ person who thinks it is so unfair that Warren Buffet pays less percentage in taxes than his secretary. I was disappointed in your answer, which does not correct the misinformation that implies that Mr Buffet pays less tax than his secretary.

First, as you noted, investment income is indeed taxed at 15%. This is in effect double taxation as the earned income that Mr. Buffet invested was taxed at his marginal rate of 36%. Comparing apples to oranges (work income vs investment income).

Second, one should not look at the percentage. In gambling terms, one should look at the ’payout’ instead. I am very certain that Mr. Buffet paid millions of dollars in taxes in the same year that his secretary paid thousands of dollars. Shouldn’t your reader be more outraged that one citizen of the country is paying 1000’s of times more than other citizens for the same government services? Once could just as easily say "I heard that Warren Buffet paid 1,000,000 times more taxes than his secretary, that is outrageous!"

Just thought I’d point out that only looking at "percentage" and not "actual payout" is a fallacy. Similar to many of your gambling fallacies.

Best Regards,

Kevin A. (Dallas)