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Ask the Wizard #241

I read with fascination the Wizard's blog about Arnold Schwarzenegger’s veto letter. My question has to do with the governor’s ridiculous but predictable response. The governor stated that it was just a ’wild coincidence’. Notwithstanding the overwhelming circumstantial evidence (The bill’s sponsor and letter’s addressee was the person who had hurled insults at the governor a week earlier), do you have an estimate of what the odds are of an exactly seven-line letter spelling this phrase by chance? I think taking into account the letters used, it will be even more improbable than just assigning a 1 in 26 chance to each. It doesn't seem like U, Y, and especially K are common word-starting letters.


If you are easily offended, please skip to the next question.

For the benefit of my readers who didn’t read that blog, look at the first letter of each line in this memo by California governor Arnold Schwarzenegger (PDF), starting with the line beginning with the letter F.

This was discussed in my companion site Wizard of Vegas. To find an answer, I found a frequency of each letter of the first word in the English Language at Wikipedia.

Word Frequency by First Letter

Letter Frequency
A 11.60%
B 4.70%
C 3.51%
D 2.67%
E 2.00%
F 3.78%
G 1.95%
H 7.23%
I 6.29%
J 0.63%
K 0.69%
L 2.71%
M 4.37%
N 2.37%
O 6.26%
P 2.55%
Q 0.17%
R 1.65%
S 7.76%
T 16.67%
U 1.49%
V 0.62%
W 6.66%
X 0.01%
Y 1.62%
Z 0.05%

To estimate the probability that Arnold’s message was indeed just a coincidence would be Prob(F) × Prob(U) × ... × prob(U) = 0.0378 × 0.0149 × 0.0351 × 0.0069 × 0.0162 × 0.0626 × 0.0149 = 1 in 486,804,391,348. This doesn’t even factor in the fact that a line break conveniently was in the place of the space between the two words.

I’d like to thank Eliot J. and Jonathan F. for their input into this solution.

Sometimes in Washington State, they don’t charge the 5% commission in pai gow poker. They make a profit from the banker’s advantage and voluntary side bets only. How does this change the odds?

Dean from Washington

With no commission, the banker has a 1.3% advantage, and all others have a 1.3% disadvantage, assuming the player follows the casino’s house way. If the player banks half the time, then the overall house edge is exactly 0%. If the player banks 1 in 7 hands, then the overall house edge is 0.93%. If the player banks 1 in 14 hands, then the overall house edge is 1.11%. If the ratio of hands that you bank is b, then the overall house edge is 1.2% - 2.4% × b.

Sometimes in Washington, the player will be required to make the Fortune side bet to have no commission. I address this rule in my Ask the Wizard column #159.

At the Buffalo Thunder casino in Santa Fe, NM, there is a progressive side bet in Ultimate Texas Hold ’Em that I have never seen before. Do you have any odds on it?

Steve from New Mexico

Thanks for the information. I address that in my page on Ultimate Texas Hold ’Em.

Do you believe players who cash in a live poker tournament should give an additional tip if they bought a "dealer’s" add-on at the beginning of the tournament? I play in a lot of small buy-in tournaments that use these add-ons, and the winners are always reminded that "tips are greatly appreciated." It seems to me that I have already tipped, even in the tournaments I don’t cash, and additional tipping just reduces whatever small edge I may have in a form of gambling that is already hard to beat (due to "vig," formats that diminish importance of skill, etc.) On the other hand, I don’t want to seem stingy. What do you suggest?

John G. from Reno

For the benefit of other readers, a "dealer’s add on" is an optional purchase of additional chips in a poker tournament. Usually the cost per chip is less for the dealer’s add on than the original entry fee, in which case buying it is a good value. To answer your question, I think you are perfectly justified in reducing the tip if you come in the money, whether or not you purchased the dealer’s add on. I would liken it to tipping at a restaurant if they already added an 18% service fee. An appropriate winner’s tip, in my opinion, is whatever the dealers would have been tipped had they been dealing cash games over the duration of the tournament, less whatever they took in from the dealer’s add on.

Let me use this opportunity to state that I oppose all additional tournament costs, unless the extra money paid goes into the prize pool, which is usually not the case. Tournaments are usually structured in a way that paying the additional fees are a good value, so most players invoke the right, including me. Your odds of winning are significantly reduced if you don’t. However, if every player pays the additional fees, then they should drop the pretenses and just charge more for the tournament in the first place.

What do you think about that Patriot’s call of going for 1st down when they are 4th down on their own 28 yard line — and they are leading by 6 points ?! What is the odds of making a first down in that situation, and what would you have done?

Tom from Hong Kong

For the benefit of other readers, this refers to a Nov 15, 2009 game in which the Patriots were up by six points over the Colts. There was 1:57 left in the fourth quarter, it was fourth down and about 1.5 yards, and the ball was on the Patriots’ 28 yard line. Patriots’ coach Bill Belichick made the now controversial decision to seal the victory and go for the first down on fourth and short, rather than kick the can down the road and punt.

There is a good column about this in the Las Vegas Review Journal. It quotes professional gambler and fellow actuary Steve Fezzik as saying that the odds favored going for the first down. I agree completely. In general, I think that other coaches punt too often, and too afraid of taking risks. To argue my position I asked fellow math head and sports bettor Joel B., who is much better than me at analyzing football odds mid-game. He offered the following odds:

  • Patriot’s probability of making the first down: 60%
  • Patriot’s probability of winning, given they make the first down: 100%
  • Patriot’s probability of winning, given they miss the first down: 50%
  • Patriot’s probability of winning, given they punt: 75%

So, the probability of winning by going for the first down is 60%×100% + 40%× 50% = 60% + 20% = 80%. That is greater than the 75% by punting.

The Monday morning quarterbacks can vilify Belichick all they want, but I applaud his decision. He shouldn’t be judged by the outcome of the game, but by whether or not the odds favored what he did at the time. I strongly feel that they did. A week later in the Ravens/Steelers game the Ravens went for it on 4th and 5, and made it. Although it was a different kind of situation, I’ve yet to hear anybody second-guess that decision.

In the interests of fairness, I am providing a link to an article taking the opposite point of view, titled Belichick’s fourth-and-reckless by Bill Simmons at