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Ask the Wizard #143

Party poker has added a side bet in Hold 'em. It pays 7 to 1 if the flop is all red or all black. (You must choose the correct color.) Is this a sucker bet, or should I be asking how bad of a sucker bet is this? Thanks for the great site.

Kerry T. from Austin, TX

Thanks for the compliment. The probability of that the flop will all be the same of a particular color is combin(26,3)/combin(52,3) = 2600/22100 = 2/17 = 11.765%. The expected return on this bet is (2/17)*7 - (15/17) = -1/17 = -5.882%.

Sir, I recently read in a book about odds that the odds of hitting all 20 numbers in keno are a quintillion to one. The book described this by saying that if there were one drawing per week and everyone on earth always bought a ticket, it would take 5 million years to produce a winner. My question is, is there a prize for hitting all 20, and if so, has anyone ever hit it? I’ve heard that no one has ever hit keno in the history of Vegas, it this true?

Tim from Greenville, SC

The probability of hitting all 20 is 1 in combin(80,20) = 3,535,316,142,212,180,000. So the odds are more like 3.5 quintillion to one. Assuming 5 billion people on earth, and they all played once a week, there would be one winner every 13.56 million years on average. Most casinos pay the same amount for hitting close to 20. For example the Las Vegas Hilton pays $20,000 for hitting 17 or more out of 20. I have never heard of anyone every hitting 20 out of 20, and doubt very much that it has ever happened.

I’m confused about your claim in the September 25, 2005 column that a suited 5/6 is the best hand to have against pocket aces in Holdem. While your program is undoubtedly right and squares with all the other programs, I am still a little puzzled as to why 5/6 is better than 6/7 (in the sense of losing less as opposed to winning more), especially when there are several obvious hands in which it is worse, namely the fact that the A234 set of straights all lose to four aces when that’s the fifth card, whereas the corresponding 2345 straight doesn’t have tis problem. There are some other anomalies I’m looking at, but what is interesting to me is that what seems like a pure logic problem is far from straightforward and requires a machine assist to guide intuition.

Jonathan F. from New York, NY

Okay, a suited 5/6 against pocket aces, both of different suits, will win 22.87% of the time, tie 0.37%, and lose 76.75%. A suited 6/7 will win 22.88%, tie 0.32%, and lose 76.80%. So the suited 6/7 will win 0.01% more. However the suited 5/6 is better because it ties 0.05% more. The reason for this seems to come down to the straights. There are going to be more waits to form a straight on the board if all the mid-cards in left in the deck. Removing a 7, as opposed to a 5, makes it more difficult to make straights with the remaining cards, thus making ties less likely, and thus the expected value less.

Which do I have better odds of winning:
A. one shot at 1 in 4
B. five shots at 1 in 20

Mike from Lansing

The probability of A is obviously 25%. The probability of getting zero shots out of five is 0.955=77.378%. So the probability of getting at least one out of five is 100%-77.378%=22.622%. So A has the higher probability.

Me and my girlfriend go to the casino often and play Pai Gow poker. I was wondering if it was statistically better for us to each play half our money, or one of us to play with it all, or are they identical odds?

Beau B from Marysville, WA

The odds are the same. However it will be less volatile for both of you to play at half the bet size.

Since you're doing football now I have a question about parlays. I recently placed a bet where I picked the over/under in each of the 4 quarters of the Steelers/Chargers MNF game and won. (Only quarters, no half or total.) Now the sports book won't pay because they say there's correlation -- that winning one quarter makes it more likely that I'll win another quarter. I believe each quarter of a game is mutually exclusive but they disagree. What does the Wizard think?

Phil from Chicago

First, whoever accepted this bet should honor it, on principle alone. A gentleman honors his debts, especially gambling debts. Second, although I haven't studied it I think the quarters may actually be negatively correlated. For example if the first quarter has a low total it may be more likely that either team will have good field position at the beginning of the second quarter, and thus likely to make the second quarter high scoring, and vise versa.

Playing Texas Hold’em with 10 players using a standard 52-card deck, after the first two cards are dealt to each player, what are the odds that the "flop" (the next three cards) will all be the same suit? Does it make a difference if my hand has both cards of the same suit and/or each one a different suit?

Mark from Milford

Before considering your own cards the probability is 4×combin(13,3)/combin(52,3) = 5.1764706%.

Another way to look at it is the probability second card in the flop will match the first in suit is (12/51). The probability third card in the flop will match it is (11/50). (12/51)×(11/50)=5.1764706%.

The odds change a little if you consider your own cards. If you have two cards of the same suit, then the probability of a suited flop is pr(flush in same suit) + pr(flush in a different suit) = combin(11,3)/combin(50,3) + 3×combin(13,3)/combin(50,3) = 5.2193878%.

If you have two cards of a different suit, then the probability of a suited flop is pr(flush in suit in common) + pr(flush in a different suit) = 2×combin(12,3)/combin(50,3) + 2×combin(13,3)/combin(50,3) = 5.1632653%.

I’ve been emailing this girl through the E Harmony system, which allows for anonymous communication, for a little over two months now. Probably over a month ago, I asked if she would like to meet in person, and she mentioned she would in "2-3 weeks." We have great conversations over email, but I must admit the lack of taking things to the next level -- as in coffee -- is starting to seem to be a red flag. That said, we’ve both been very busy over this time. What do you think are the three most probable reasons for her reluctance to get together in person? I forgot to mention that she’s a psychiatrist, so of course I could be the victim of some grand experiment. That was a joke, but let me know what you think. Thanks!

Tim W from Cleveland

She might be using you as her psychiatrist. It sounds like she is leading you on and just wants somebody to listen to her. Another possibility is she is in another relationship that is rocky and you are her backup plan. You can’t waste your time in limbo indefinitely. I would tell her to not contact you until it is with a meeting date.