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Ask the Wizard #145

My question is about a problem that is known as the "two envelope paradox". You are on a game show. In front of you are 2 envelopes, each containing an unknown amount of cash. You are told that 1 envelope has twice as much money as the other. You are now asked to choose an envelope. You choose one. It contains $50,000. Now you are told that you can keep the envelope you picked, or swap for the other one. Should you swap? Knowing ahead of time that you could swap, then it doesn’t matter, as you would just choose the envelope you ultimately want. But because you only find out about swapping after you choose an envelope, then the original selection and the option to swap are 2 independent events, correct? That said, when deciding to swap or not, the other envelope contains either twice as much or half as much as what you currently have. So it has either $100K or $25k. Since there is a 50% chance of either occurring, the Expected Value of the other envelope is $62,500. Generically speaking, if we let x = the amount you originally selected, then the other envelope's EV is 1.25x. Therefore it is always correct to swap. Is this correct? Thank you.

Derek from Boston

I'm very familiar with this problem. I address it on my web site of math problems, problem number 6. There I address the general case, including not looking in the first envelope at all. However to answer your question we can not ignore the venue of where the game is taking place. You said it was a "game show." On most game shows $50,000 is a nice win. Few contestants on the Price is Right ever make it that high. I would guess that fewer than 50% of players on Who Wants to be a Millionaire get that high. Meanwhile wins of $25,000 are not unusual on game shows. Cars are won routinely on the Price is Right, which have values of about $25,000. The $32,000 level is a common win on Who Wants to be a Millionaire. The average win on Jeopardy per show is roughly $25,000. The great Ken Jennings averaged only $34,091 over his 74 wins. So, my point is that $50,000 is a nice win for a game show, and $100,000 wins are seen much less often that $25,000. Thus as a game show connoisseur it is my opinion that the other envelope is more likely to have $25,000 than $100,000. So I say in your example it is better to keep the $50,000. It also goes to show you can never assume the chances that the other envelope has half as much or twice as much are exactly 50/50. Once you see the amount and put it in the context of the venue it is being played you can make an intelligent decision on switching, which throws the 1.25x argument out the window.

Do you advise playing the money line or the spread on your NFL picks or does it not matter one way or another?

Aaron Kelly from New York

Regardless of the reason for making the bet, in general it is better to bet underdogs on the money line and favorites against the spread.

There’s a variant of 5 card stud pokercalled Soko. It plays just like regular poker, except that there are two additional hand rankings. Above a pair is a 4 card straight, then a four card flush, then two pair. The rankings then proceed normally. Where should a 4 card straight flush rank if added as a hand ranking?

Nathan from Tuscon

The number of ways to make a 4-card straight flush is 4*(9*46 + 2*47) = 2032. There are 3744 ways to make a full house and 624 ways to make a four of a kind. So the four-card straight flush should fall between a full house and four of a kind.

If an immediate family member is a compulsive gambler, owes everyone money, and can’t function in the real world, is there a "blacklist" or other device on which to place his name so he will be banned from using internet gambling sites?

Barbara from Englewood

There should be but I have never heard of such a list. Even if there is such a list I think he would have to put his name on himself. The more respectable Internet casinos honor their own lists, and I have heard of loss refunds if the gambler proves he is getting treatment.

At our (draw) poker game a player held a high card "kicker" to improve his pair on the draw. That is counter-intuitive to me. Does holding a kicker improve your chances of improving a pair (5 card draw poker)?

Jim from Albuquerque, NM

If you hold only the low pair then the probability of improving the hand to two pair or better is 28.714%. If you hold the pair and a kicker the probability of improving to a two pair or better is 25.902%. So the probability of improving to a two pair or better is higher by holding the pair only. However if you assume that you’ll need a high two pair or better to win then the probability of achieving that will likely be higher holding the kicker, depending on the specific cards and how you define "high."

When I play online poker, are the cards 100% set (as they would be with a real deck of cards) when the hands begin or does the RNG keep ’spinning’ before each card is dealt and thus each card is random?

Andy from Indianapolis

Each card is random whether the RNG is ’spinning’ before each card is dealt or not. As for whether the RNG keeps spinning, I don’t know, but mathematically it doesn’t make any difference.

What happens with accrued money in a progressive slot jackpot that is not won at the time that the progressive is discontinued? Seems to me that the accrued amount over the base should be allocated back to the players in some way since it was part of the return.

Roger from Dallas, Texas

According to Nevada Gaming Control Board regulation 5.110.5(c), the casino licensee must add the progressive jackpot to a similar game at the same establishment.

I have always wanted to do some of my own NFL handicapping but I am having a hard time finding a site, hopefully free, to download historical data by team. Any suggestions? Downloadable files would be my preference since cutting and pasting off of a web page isn’t real practical but I would be willing to do so if needed. Also, information such as weather and turf conditions would be a great help.

Ed from Indianapolis

Personally, I use the NFL Access database from Mr. NFL, which costs $99. If there is anything as good for less I'm not aware of it.

What are the odds of a pair showing on the board on the flop in Holdem? ie., A A 10 or 5 Q 5, etc.

Stephen from Addison

13*12*combin(4,2)*4/combin(52,3) = 3744/22100 = 16.941%.

Many of your responses to questions about slots refer to e-prom which you say is regulated and requires approval by a state agency prior to changing. However, in California I know of no regulatory agency that requires Indian casinos to submit changes for approval prior to changing any EPROMs. I assume Indian casinos can make the changes whenever they desire (yes/no?).

Fred from Bonita

When I write about government regulations I almost always am talking about Nevada. Many other jurisdictions more or less mirror Nevada laws. However Indian casinos are largely self-regulating. As far as I know they can change EPROM chips at will and not answer to anybody about it.

I remember that if 22 people are in a room the odds are even that two will celebrate the same birthday [month and day, not year]. I have forgotten how to do the math to prove this. Could you please provide it.

Dean from Bainbridge Island, WA

I think I have answered this before but the 50/50 point is closer to 23. To make things simple let’s ignore leap years. The long answer is to order the 23 people somehow. The probability that person #2 has a different birthday from person #1 is 364/365. The probability person #3 has a different birthday from persons #1 and #2, assuming they are different from each other, is 363/365. Keep repeating until person 23. The probability is thus (364/365)*(363/365)*...*(343/365) = 49.2703%. So the probability of no match is 49.27% and of at least one match is 50.73%. Another solution is the number of permutations of 23 different birthdays divided by the total number of ways to pick 23 random numbers from 1 to 365, which is permut(365,23)/36523 = 42,200,819,302,092,400,000,000,000,000,000,000,000,000,000,000,000,000,000,000 / 85,651,679,353,150,300,000,000,000,000,000,000,000,000,000,000,000,000,000,000 = 49.27%.

First, thank you for the great site. I went to Las Vegas for this first time this past summer and I played double-deck blackjack at the Orleans. I noticed that after a dealer shuffled both decks, the dealer asked the player to cut the deck. Most players refused. I did not mind, so I cut the deck. Is there a blackjack cut superstition that I am not aware of, or is there a better reason why?

Sam from Price

I would say about 1/3 to 1/2 of players would at least initially decline to cut. However if everyone initially declines somebody has to rise to the occasion and do it. Sometimes when players who refuse to cut will say something like "I don’t want the blame for a bad shoe" or "I’m unlucky." I’ve never seen it put into words but there does seem to be a superstition that the cut is critical to the flow of the shoe, and thus the act should only be done by a competent cutter. Of course this is nonsense. For recreational play it doesn’t make any difference whom cuts or where they cut.