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Ask the Wizard #161
Jon from Lafayette, CO
Let’s examine the general case first.
Define p as the probability that the next four of a kind will be one that you need for the promotion.
Define q as 1 - p.
Define m as the expected number of four of a kinds to get one that you need.
The sum of probabilities is 1. Thus,
(1) p + p×q1 + p×q2 + p×q3 + p×q4 + ... = 1
The following is the formula for m in terms of p and q.
(2) m = 1×p + 2×q×p1 + 3×q2×p + 4×q3×p + 5×q4×p + ...
Multiply both sides of (2) by q.
(3) mq = 1×pq + 2×p×q2 + 3×p×q3 + 4×p×q4 + 5×p×q5
Subtract (3) from (2)
(4) m - mq = p + pq + pq2 + pq3 + pq4 + ...
The right side of (4) equals 1 from (1).
(5) m - mq = 1
(6) m×(1-q) = 1
(7) m = 1/(1-q) = 1/p.
So, if the probability of an event is p, then on average it will take 1/p trials to occur.
To get back to the problem at hand, it will obviously only take one four of a kind to cross the first one off the list. The probability the next four of a kind will be one that you need is 12/13. So, on average, it will take 13/12=1.0833 trials to get it. Once you have two crossed off the list, the probability the next one will be one that you need is 11/13, so that will take 13/11=1.1818 more trials to get the third one.
Following this pattern the total expected number of four of a kinds to get at least one of each kind is
1 + (13/12) + (13/11) + (13/10) + ... + (13/1) = 41.34173882.
My webmaster Michael Bluejay insisted on providing his own answer first:
Actually, drawing lots wasn’t their first idea for deciding who got Jesus’ robe. First they were going to draw a horse, but they didn’t have the right color crayons. So they each decided to draw whatever they wanted, and one drew Carey while another drew Barrymore. But the judging became an apples-to-oranges kind of comparison so they settled on drawing lots. Of course, everyone likes lots because lots means plentiful. You always see signs that say "Lots for Sale", but you never see a sign that says "Only a Little Bit for Sale". When you think about it, that place "Big Lots" is kind of redundant. It’s like saying "Abundant Abundance". But if companies can get away with saying "Pizza Pizza" (or agar agar) then I guess there’s no problem. Anyway, to answer your question, lots were first used for gambling in Cow Bingo. You know, it’s the game where a cow is placed on a lot marked off in a grid and people bet on which grid square the cow will poop in. The most famous use of lots in gambling is their role as the first part of the LOTtery.
(groan) Now that that’s over with, I asked my friend and bible expert, Tom R. the "Watchman on the Wall", about this. He quoted various bible dictionaries. The bottom line is that lots were not used for gambling but to choose a name randomly. This was accomplished by writing one name each on pieces of wood or stone, putting them in a bottle, and shaking just one out.
Linda from Atlantic City
I forwarded this story to Brian, who is a former gaming regulator and current operator. Here is what he wrote.
All of the table limit signs usually have the caveat "management decision is final" - not much comfort to the player, but they’ll fall back on this for justification. In the scenario described, I would have allowed the hand to continue especially if all of the cards were already out. If I had concerns, I would change the deck out after the hand. Many casinos won’t allow 3CP players to even look at their hands until all cards are dealt. This was cutting into my hands per hour so I changed the procedures. Since the potential appeasement payout for a person that receives a good hand and then the shuffler dies is relatively small, I’m willing to take the risk. In Caribbean Stud, no one touches the cards until they are all on the table.
Kevin from Golden, CO
There are 63 total games (32+16+8+4+2+1). Each game has two possible outcomes. So the total number of ways the tournament could play out is 263 = 9,223,372,036,854,780,000.
Brad from Las Vegas
I’m afraid I don’t know much about college basketball. However, I agree that gamblers prefer to get odds rather than lay them. Nevertheless, I still say that in the NFL square money usually falls behind the favorite. For this reason, in any given Super Bowl, the spread will not be in synch with the Money Line. As an example, the 2005 Super Bowl had a 7-point spread. Normally the money line on a 7-point favorite is -300. However, on New England it was around -250. My explanation is that Eagles fans were disproportionately betting the money line, while New England fans were giving up the 7 points, creating value for New England on the Money Line.
Chris K from Los Angeles
You could do a tournament. Every player will buy in for the same amount of non-cashable chips. Establish somebody to be the banker, paying off bets as in normal craps. Whoever has the most chips after some benchmark, for example x 7-outs, wins the pool. Since you will have an even chance with everybody else, I think it would be okay to ask for tips for the use of your house.
Sophia from Berkshire
What did you do to this guy to make him switch sides? My advice is to put your cards on the table and tell him your concerns. If he chooses not to confide in you, perhaps a time apart would help him to sort things out. I’m not big on staying in a state of limbo in a relationship, you should either be going forward or getting out.
Brittney from Suitland
No. As I’ve said before, I’m not that big on forgiveness when it comes to cheating. Once is optional, twice and you’re a fool, three times and you are a hopeless door mat.
Amy from Jacksonville
One thing I repeat a lot is that once you end a relationship, truly end it. It is easy to say this in retrospect but seeing him was a big mistake. You were only giving him false hope. Tell him in no uncertain terms that you two are finished and he is never to contact you again. Also, nothing is ever meant to be. You are the master of your own fate. Apologize to your current boyfriend and ask him to help enforce your new no-contact rule, with force if necessary.
There are three women at the casino where I work (we’ll label them B, C, and J). B and C are both 23 years old. J is 26 years old. I will be 40 in June. B and J work mostly graveyard shifts (as do I). C works daytime. If I had to choose I would go for C (but I rarely see her). But me and B were holding hands one night (even though we were drunk). I flirt like crazy with J. So based on all these variables, which woman would I have the most success with?
Jason from Vancouver, BC
Thanks! I’m not big on large age differences. Based on the information given I favor J. This is because of the lesser age difference, common work schedule, and the flirting shows you like her. All things being equal, any one of them could be the best for you. However, my advice is to be practical and go with the known and convenient choice.
Alex from Taylor, TX
p.s. After publication of my original answer I received the following e-mail about the dangers of sodium thiopental. The link I provided talks of common usages for euthanasia and lethal injection, which I thought would scare off anybody who didn’t realize I was joking. However lest anybody just rush out and buy some I will post his e-mail. He suggested I change "sodium thiopental" to "truth serum" but I would rather my audience learn something about chemistry.
I love your site. It’s fabulous. I have a couple of advanced degrees, including one in biostatistics, and I could never program a computer to generate all the odds as wonderfully and cleanly as you do, nor could I explain them in such clear terms. You are unquestionably an authoritative expert when it comes to gambling.
I’m not just shining you on, though - I do have a point. In your April, 2006 advice column someone asked you how he could tell if his girlfriend was cheating. You replied "Sodium thiopental." I laughed. It is a funny joke.
Unfortunately, however, even a brief perusal of your site is enough to convince anyone that you are an authoritative expert. I am concerned that someone who might be a bit less sophisticated than the median might take your advice seriously, obtain some sodium thiopental, and administer it to an unwitting victim.
Thiopental’s really dangerous. Its therapeutic-to-toxic ratio is roughly 1:2, which means it has to be carefully dosed by weight. Physicians like me don’t even use it anymore, preferring the newer class of drugs called the benzodiazepines which have a therapeutic-to- toxic ratio of closer to 1:10. And its interaction with alcohol is dangerous and unpredictable. Frankly, I’d much rather do my gambling with dice.
I wonder if you might consider changing your answer to something similarly amusing but possibly more harmless, such as "Truth serum?"
Dave F., MD