Ask The Wizard #156

For my readers who may not understand the question, a Super Bowl pool has a 10 by 10 grid. Players will buy squares for a set amount each, writing the name of the buyer in each square. After all 100 squares have been purchased the rows and columns should be given headings randomly from 0 to 9. Kind of like a 10 by 10 multiplication table, only with the row and column headings randomly shuffled. Then the terminal digit of each team’s score will used to determine the winner. The reason for the shuffling is that some terminal digits are much more likely than others, as you will see below. For example, whoever ended up with the Seattle 0, Pittsburgh 1 square would have won the pool, because the final score was Seattle 10, Pittsburgh 21.

The following table shows the frequency of each combination in every NFL game from the 1983 to 2005 seasons. It should be noted that the 2-point conversion rule came along about 1998, which would have the effect of smoothing out the distribution a bit.

The next table shows the probability of each combination based on the totals in the table above.

Although there is a ceremonial home team in the Super Bowl, I think we can ignore that. Let's also ignore everything about both teams and go strictly off of the historical averages above. Finally, let’s ignore the fact that the Super Bowl can not end in a tie, which would make a match like 4/4 a little less likely to win. So let's take an average when the terminal digits are different. For example the probability of your Pittsburgh 7, Seattle 4 square would be the average of Away 7, Home 4; and Away 4, Home 7. Doing so results in the following probability for each of your squares.

Pitts 7, Sea 4: (0.0346+0.0381)/2 = 0.0364
Pitts 7, Sea 0: (0.0385+0.0420)/2 = 0.0403
Pitts 4, Sea 4: 0.0192
Pitts 4, Sea 0: (0.0216+0.0245)/2 = 0.0231

So your total probability of winning one of these is 11.90%. Considering you only covered 4% of the squares you made out well.

Although you didn't specifically ask, here is how often each terminal digit occurred. It shows that overall from most the frequent the order is 7043168952.

Finally, here is the probability of each terminal digit.

For more on this topic, visit Football Squares Mathematical and Statistical Strategy.

If a royal flush paid the same as a straight flush then 9/6 Jacks or Better would have a return of 98.03%.

1 in 635,241.

I get asked variations of this question a lot. The fact of the matter is I have written hundreds of different simulations. I write the simulations myself in C++ to do exactly what I want. Those writing usually seem to be looking for something to test betting systems. I’m afraid I have nothing and know of nothing that lets the user input how the betting system works and then tests it. If there were something that worked perfectly what you would learn from it is that all betting systems are equally worthless, exactly what I have been saying for years.

For the benefit of my readers, table 3 pays $500 for a four of a kind, $100 for a full house, and $50 for a flush. The breakeven meter on the Northern Indiana side bet pay table is $369,290.00

In your example the casino’s hold is 75%. That is the ratio of how the value of chips players walk away with to the amount of money the dealer drops in the box. Although there are historical averages there is no way to go from house edge to hold. The hold depends in part on how long the player will circulate through his chips, and there is no way to know that.

For my readers who may not know, a hand of Omaha has nine cards. If the player is allowed to use any nine cards the probability would be (13*combin(48,5)-combin(13,2)*44)/combin(52,9) = 0.00605. However, if the player is forced to use exactly two of his four hole cards the probability is

(13*combin(4,2)*combin(48,2)*combin(2,2)*combin(46,3)-combin(13,2)*combin(4,2)*combin(4,2)*combin(2,2)*combin(2,2)*44)/(combin(52,4)*combin(48,5)) = 0.00288

Note that these formulas adjust for the possibility of getting two four of a kinds.

I’m afraid I take his side. He wouldn’t fault you for keeping old love letters from ex-boyfriends would he? However I don’t blame you for not wanting to be a part of the collection. It may make you feel better if he says he goes along with your request, however I tend to think he’ll conveniently forget about a backup somewhere. So I don’t think your ultimatum is going to solve the problem. If you can’t accept his uninhibited attitude then I think, indeed, the relationship won’t last.

For Las Vegas why not use my own blackjack survey at the Wizard of Vegas. For the rest of the United States and Canada, I highly recommend Stanford Wong's Current Blackjack News. That will tell you the exact blackjack rules at almost every casino in both countries.

It could be done either way. I think most slot makers prefer to have the prizes constantly mixed up, so the outcome is a function of both which box you pick and when you pick it. Mathematically speaking it doesn’t make any difference which way of the two they program it.

I was convinced he was cheating by the time you got to the underwear part. Wake up and smell the coffee.