Ask The Wizard #156

I am in an NFL squares pool and I have drawn very good numbers. 7 and 4 for pittsburgh and 4 and 0 for seattle. I was wondering what the probability is on me winning this year?


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.

Terminal Digit in the NFL - Away by Home Total

Away Team Home Team Total
0 1 2 3 4 5 6 7 8 9
0 126 104 34 160 138 37 99 237 64 32 1031
1 73 40 17 41 103 21 36 117 67 31 546
2 25 15 1 20 30 9 13 33 9 12 167
3 194 69 33 66 102 40 102 132 43 40 821
4 122 133 27 78 108 27 48 215 59 35 852
5 32 12 12 21 21 9 13 36 22 5 183
6 91 47 17 75 57 7 28 57 25 39 443
7 217 115 35 135 195 47 65 125 69 47 1050
8 43 59 12 24 41 23 20 38 15 7 282
9 48 28 17 33 40 14 29 33 14 7 263
Total 971 622 205 653 835 234 453 1023 387 255 5638

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

Terminal Digit in the NFL — Away by Home Probability

Away Team Home Team Total
0 1 2 3 4 5 6 7 8 9
0 0.0223 0.0184 0.006 0.0284 0.0245 0.0066 0.0176 0.042 0.0114 0.0057 0.1829
1 0.0129 0.0071 0.003 0.0073 0.0183 0.0037 0.0064 0.0208 0.0119 0.0055 0.0968
2 0.0044 0.0027 0.0002 0.0035 0.0053 0.0016 0.0023 0.0059 0.0016 0.0021 0.0296
3 0.0344 0.0122 0.0059 0.0117 0.0181 0.0071 0.0181 0.0234 0.0076 0.0071 0.1456
4 0.0216 0.0236 0.0048 0.0138 0.0192 0.0048 0.0085 0.0381 0.0105 0.0062 0.1511
5 0.0057 0.0021 0.0021 0.0037 0.0037 0.0016 0.0023 0.0064 0.0039 0.0009 0.0325
6 0.0161 0.0083 0.003 0.0133 0.0101 0.0012 0.005 0.0101 0.0044 0.0069 0.0786
7 0.0385 0.0204 0.0062 0.0239 0.0346 0.0083 0.0115 0.0222 0.0122 0.0083 0.1862
8 0.0076 0.0105 0.0021 0.0043 0.0073 0.0041 0.0035 0.0067 0.0027 0.0012 0.05
9 0.0085 0.005 0.003 0.0059 0.0071 0.0025 0.0051 0.0059 0.0025 0.0012 0.0466
Total 0.1722 0.1103 0.0364 0.1158 0.1481 0.0415 0.0803 0.1814 0.0686 0.0452 1

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.

Terminal Digit in the NFL — Away and Home Totals

Away Team Home Team Total
0 1031 971 2002
1 546 622 1168
2 167 205 372
3 821 653 1474
4 852 835 1687
5 183 234 417
6 443 453 896
7 1050 1023 2073
8 282 387 669
9 263 255 518

Finally, here is the probability of each terminal digit.

Terminal Digit in the NFL — Away and Home Probabilities

Away Team Home Team Total
0 0.1829 0.1722 0.1775
1 0.0968 0.1103 0.1036
2 0.0296 0.0364 0.033
3 0.1456 0.1158 0.1307
4 0.1511 0.1481 0.1496
5 0.0325 0.0415 0.037
6 0.0786 0.0803 0.0795
7 0.1862 0.1814 0.1838
8 0.05 0.0686 0.0593
9 0.0466 0.0452 0.0459

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

Understanding that optimal Jacks or Better play yields 99.54% return on a 9/6 machine, I have a quick question: If there were no such thing as a Royal Flush, but everything else remained exactly the same, what would the return be on the same machine? Thank you.

Michael from Seattle

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

There are 75 multiple choice questions in an exam. Each question contains 4 possible answers only 1 is correct. The exam pass mark is 50%. What are the chances of passing the exam by guessing each answer?

Wendy from London

1 in 635,241.

Just wondering where I might obtain a good testing program like the billion-hand simulator you often refer to. Thank you.

Michael from Los Angeles

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.

I would like to find out what the breakeven point is for the Caribbean Stud Progressive Jackpot Side Bet for a payoff table that I saw in Northern Indiana that is not included on your Caribbean Stud page. The table I am referring to is the same as your Table 3 except that the payoff for a straight flush is a fixed $5,000 instead of 10% of the jackpot. I saw it at Blue Chip Casino in Michigan City, Indiana, and was told by other players that that is the standard table in Northern Indiana (in Southern Indiana, your Table 3 is standard). Thanks much for the wealth of information you provide on your site.

Kevin from Cincinnati

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

When I enter a casino with $100 and play roulette, I don’t expect to leave it with 4% less but with 25% less. Does this phenomena has a formal name? (I was googling "real house edge") Is there any formula/model to predict this "real house edge"? Which games are better and which are worse in this parameter?

Jacob from Tel Aviv

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.

What is the probability of getting a four of a kind in Omaha?

Kevin from Calgary, Alberta

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.

The other day i was on the computer while my boyfriend was out,and stumbled across so really horrible porn photos of him and his ex-lovers,altho i shouldnt have been looking at those folders,i decided that i should tell him what i did and tell him how sick i felt that he has them.he then dropped by accident that there are videos too.I told him how i feel about him having that stuff(ie him having sex with his ex’s),but he doesnt see anything wrong with it-he says theyre a souvenir of the past(altho he also says that he didnt love them-so what is it a souvenir of?).i never had let anyother guy take photos of me naked, but now i really feel used and abused!-i said he could either keep the others and delete mine or visa versa-but he wont do either.Because of the age gap between us, he just calls me immature-i dont think im being unreasonable-as i said i feel used, and frankly i dont know whats going to happen between us? can our relationship last? and am i really in the wrong? (i did afterall really really appologize about looking at the folders)

Lucia from Geneva

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.

How can I get a directory or which casinos (worldwide) offer double deck "pitch" blackjack? I have scoured the internet and even most of the casino's websites do not specify.

Kim from Novi

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.

On slot machines with bonus rounds like "Jackpot Party" or the one with the leprechauns are the winning boxes or pots of gold predetermined the second you hit the bonus round or on an individual basis by a random number generator when you select each box.

Jerry from Tacoma

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.

Help! I think my boyfriend of over a year is cheating on me. Lately he has been getting a lot of instant messages from his ex. They haven’t spoken for a long time, and he told me it was for closure. Then one morning he called me her name, twice. I was upset, but he said he must have been dreaming so I forgave him. A week later when I was doing his laundry I found another girl’s underwear in his dirty clothes, he said it was his sisters, but she denied ever having owned such a color and it is the wrong size for her. I started getting suspicious and one night when we got into a fight over it, he tried to calm my suspiciousions by telling me that, ’yea, i really wanted to get with her. I want to show her what she is missing. And it would be so easy, you would never find out," . . . then he went on to tell me that he didn’t though, because he knew that would mean the end of us. It is a month later and he is still acting strange. He used to always answer his phone, and now he doesn’t get back to me for hours sometimes, he is always ’leaving his phone in the car’. I have known him for four years, and he carries his phone almost everywhere, and ALWAYS answers when people call. Also there have been a few times when he has gone ’out with his cousin’ that he was supposed to be in by 1am, and when I look at the clock as he is creeping in, its 5am. He always has some great reason, but I am continuously forced to wonder if it is getting to be too much, and I am just a fool.

A Jersey Fool?

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