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Ask the Wizard #145
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.
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.
Nathan from Tuscon
The number of ways to make a 4card 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 fourcard straight flush should fall between a full house and four of a kind.
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.
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."
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.
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.
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.
Stephen from Addison
13*12*combin(4,2)*4/combin(52,3) = 3744/22100 = 16.941%.
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 selfregulating. As far as I know they can change EPROM chips at will and not answer to anybody about 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)/365^{23} = 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%.
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.