Share this

Ask the Wizard #161

A casino has a promotion where you get a bonus for getting each of the different 4-of-a-kinds in video poker. For simplicity, I assume each 4 of a kind occurs with equal probability. How does one calculate the average number of 4-of-a-kinds one must get before one can expect to have received each of the 13 different kinds at least once? Thanks very much, I really appreciate all of the information on your site!

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.

Lots are mentioned several times in the bible, most famously for using lots to decide who went home with Jesus’ robe. What exactly are lots and were they used for gambling?


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.

Last night, a fellow player at a 3-Card Poker table dropped his cards on the floor. After the cards were dealt, he had picked them up and was opening them up with one hand while showing them to his son who was a "spectator" behind him. Somehow he fumbled them and all 3 fell to the floor. I had already looked at my two hands ( a flush and a straight!) and placed my bets. The floorman told everyone all their hands were "dead". When I asked for the supervisor, he came over, thought about it for a few seconds, and also agreed it was a "dead" hand. Is this the standard rule on this type of occurance? Should all hands have been disqualified, or just the offender’s (who, by the way, had a Q,6,5). I was betting $10 per spot, so my loss was $150 in winnings (I am assuming that I beat the dealer - he never did show his cards). Should I have asked for a copy of the written rule on this? Should I have not given in after the supervisor’s ruling?

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.

How many possible combinations are there for the NCAA March Madness tournament. In other words had I wanted to cover every possible outcome for the 64 team tournament, how many differenct combinations would I have to cover. I have seen one answer that says it is 2 to the power of 64 which is 18.4 quintrillion. This number seems too large to me. I am a math major but don’t have my old probability book handy to figure out the combinations/permutations, so I thought I would ask the Wizard.

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.

In your answer to Edward from Placentia you say, "Square action on the money line tends to favor the favorite, creating value on the underdog." I’m not so sure this is usually the case. As you know, most small time gamblers like to risk a little to win a lot. This is achieved by betting the moneyline on the dog in these high profile games. While you were correct in your analyzation of the Super Bowl, I would urge you to look at this past NCAA Football championship game between USC and Texas. Tons of public money came in on Texas ML, while the spread stayed around 7. On gameday, both USC -185 and Texas +206 was available. I’m not really looking for an answer here, but I thought it might be something else for you to consider when betting sports in the future. Thanks for the great site!

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.

Do you have any good rules/setups for playing Craps at home for actual cash. I understand that to keep things legal, I can’t take a ’house cut’, but assuming a buy-in is enforced (like playing poker at home) is there a good system to play privately for money without becoming "the house" and paying winners out of my own pocket?

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.

Lately my boyfriend of 5 months has been acting distant from me. He doesn’t show me affection like he used to and he doesn’t kiss me the way he used to. Also last night when we were making love he shouted out the name of another person whom happens to be a male co-worker. Should I be concerned that he has feelings for this other man or am I over reacting? Do you think it is possible that he may be gay and is just using me as a cover?

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.

i meet my boyfriend at church, last year then we started dating for 6 months. i thought that he was going to be faithful to me but what do you know he cheated, i was looking through his phone and seen some nasty text messages. so before i could break up with him he broke up with me. then about 8 months later he said that he changed and that he want to be with me again. so i took him back. turns out that he didnt change after all. infact he was planing on dumbing me for a new girl at the church, and so he did but he dont know that i know about his plan. if he askes for a third chance should i? my heart is still with him.

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.

About 7 months ago I left my ex-boyfriend for my current boyfriend now. I love my new boyfriend with all my heart and I would do anything to keep him in my life. But my ex-boyfriend keeps calling and asking me to be with him. So feeling bad that I left him for another guy I decided to have lunch with him. Then when he took me home he kissed me. My question is if me and him are suppose to be or if me and my new boyfriend will overcome this. Thanx!

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.

Hello Wizard of Relationships. I think you should do a website devoted to relationships. So far I’ve been quite impressed with your advice. OK here goes...

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.

I know my girlfriend is cheating .how can i get her to tell me?

Alex from Taylor, TX

Sodium thiopental.

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?"

Respectfully yours,

Dave F., MD