Math/Gambling

9. The Decision Favored to Be Wrong May Be Right. That can happen if that alternative has a much larger upside, a much smaller downside, or if it can be reversed if going poorly while the other decision(s) can’t.

17. Math Questions That Need Almost No Math. Two Examples: #1. We flip coins for a dollar a flip. I start with $2, and you start with one. Algebra tells us that I will bust you twice as often as you bust me. Pure logic points out that this must be so since fair bets must break even if we do this every day. #2. I flip then you then me until one of us gets a head. Since I’m going first, math types make me 1/2 + 1/8 + 1/32……Smarter people realize that on the first round I’m 1/2 and your’e 1/4 and that ratio remains on all rounds until there is a winner. We both get 2/3.

23. If You Know the Dealer Has 17 Don’t Double Down on 11. Even though that decision will cost you a bet seven times out of 12. Because the other decision is sometimes three bets better than doubling and thus even more profitable. The general principle is that in both gambling and real life, it is not enough to see if a certain decision is “plus EV”. (Similar to #19 above.)
24. Giving Him Two Tens Which He Must Split Against Your Nine Showing. If you think this blackjack proposition favors you, read #23 again. Splitting tens against a nine is a terrible play. But only compared to the alternative of standing. To quickly realize this, you need only remember that Basic Strategy says to double down with ten against a nine even though you can’t take another card.

27. The Game Theory Solution to the Game of “Chicken”. Not my original idea but worth repeating. Two cars drive towards each other and the first one to swerve loses. In this obviously hypothetical game, the obviously hypothetical solution is to throw your steering wheel out the window, making sure the other guy sees it. There are lots of real-world analogies.
28. Well Thought Out Hypotheticals Can Be Useful. They don’t have to coincide exactly with the real-world scenario they are shining a light on. Especially if the differences between the hypothetical and what it is modelling are recognized and adjusted for.

30. It’s Worth Going to Some Lengths to Prevent Bluffs from Those Who Would Normally Play Close to Game Theory Strategy in Big Bet Poker Games. That often means making a play on the next to last round that would normally be wrong but becomes clearly right if it switches his last round bluffing frequency to a negligible percentage. Obviously, I will elaborate in the forthcoming elaboration section.
31. If 95% Repayors Are Profitable It’s Still Wrong to Relax Requirements If Your Results Are 97%. Not my original idea but an important one. A loan officer who is getting paid back 97% of the time can probably assume that the riskier customers are already 95% or less. Bank officers usually know this, but others to whom this concept applies, often don’t.

34. When Are You So Old That You Can’t Bet You Will Live Another Year? Most people guess about 90. So here is some good news and some bad news. The good news is that for typical Americans, the answer is about 111. The bad news is that 90 is not only a horribly wrong answer but also a horribly dumb one. If 90 years old is even money than the number of 100 year olds would be way smaller than 1/1024 of the 90 years old number which is clearly not true.

42. The Simple Math That Proves “Crapless” Craps is Worse. 2,3, and 12 are no longer automatic losers. Eleven is no longer an automatic winner. They are all “points”. Many think that a pass line bettor is given more than is taken away. But 2 and 12 only move up from zero to 1/7. 3 goes from 0 to 25%. Meanwhile 11 moves down from 100% to only 25%. Be on the lookout for analogies.
43. Big Favorites Are Big Underdogs to Go on Long Streaks. For instance, it’s five times easier for an 80% shot to go 19 for 20 than 20 for 20. That’s why John Wooden is probably the best coach of all time despite having the best talent. And why no one will ever top Joe Dimaggio’s hitting streak.
44. Paying for Info. Whether it be actual money or perhaps a normally negative EV play early in a poker hand that will make your opponent give away his hand, you should have an idea of the correct price. It is the weighted average of the various gains you achieve compared to your EV had you not known anything. Including the times it is zero because the info doesn’t change your decision.
45. A Positive Test for a Very Rare Disease is Probably Wrong. Not my original thought plus there are exceptions. Still, it’s something that should be more well known as are many other real-world facts that are deducible with the help of something called Bayes’ Theorem.

47. Passing 95% Confidence Criteria Doesn’t Make You 95%. An extremely serious error made even by “experts”, that a few people are trying to explain to the world. If a drug on trial beats the 95% criteria, it means the older drug or placebo tested against it did so much worse that there is only a 5% probability that the trial would come out this way if in fact the tested drug was no better. But that is NOT AT ALL the same thing as saying that there is only a 5% chance that the tested drug doesn’t work. On average its chances are much worse.
48. An Interesting Way to Beat A “No Skill No Edge” Lottery Type Game. Say I conduct a lottery where tomorrow I randomly choose a number between 1 and 100 and split the money from the one dollar tickets I sold among those who chose the winning number (refunds if no winner). Random tickets break even. But someone who pays $100 for 100 different numbers has an edge (unless everyone else does)! To see why suppose only one other player plays and buys one ticket. You win one dollar 99 times and lose $49.50 one time. The explanation is that you making sure you don’t duplicate, makes your tickets not random.
49. #48 Means More Horse Race Pick Sixes Than Expected Are Beatable. But only if you have a few hundred thousand dollars to invest in different reasonable combinations. (This was another piece of good advice I gave Jeff Yass years after he asked about 8 card stud.) Unfortunately, the opposite is true if you only are betting a few grand and you know a syndicate is betting also. (Stanford Wong got this wrong in his book.)

54. Insane EV Ignorance/Risk Aversion in Average People. We had a year-long cheap buy-in slot tournament that culminated in four finalists shooting for prizes of 10K, 20K, 50K and a million dollars on the final round. They agreed to settle for 60K each even though playing gave them an EV of 270K! The fact that the settlement was slightly better than 2nd prize swamped everything else in their minds. I will leave it to the reader to come up with ways to take advantage of this syndrome.

58. Keno, Roulette, And Even Lotteries Can Sometimes Be Better Bets than the Craps Pass Line. Even though the house edge on craps is 1.4% while its 5.5% on roulette, 25% on keno and 50% on lotteries. The basic reason is that the “worse” bets can instantly turn your one dollar into $36, $10,000, or mega millions. For craps to do that means you have parlayed a win many times and put out far more than one dollar from which 1.4% is taken.
59. Internet Poker Idea #1. After about 80% of the tournament entries are eliminated, the computer distributes half of the total prize money to the remaining players in proportion to their stacks. The tournament then continues as normal for the rest of the money. A tournament of that nature has the advantage of discouraging “nitty” survival strategies.
60. Internet Poker Idea #2 This idea and the previous one are distinguished by the fact that live poker would be hard pressed to duplicate them. (And I have several more that I am withholding for the time being.) Hi-Lo split games that award 60% to the best high hand. No longer would the admonition “Don’t play purely high hands” be generally good advice. This game, whether it be stud or Omaha, would likely result in a lot of action.

62. Deviate from Poker Early Round Game Theory Recommendations If You Are Unwilling to Carry Them Through to the End. Again, to be expounded upon later. Basically, it means that you should call fewer preflop raises than “GTO” says if you don’t want to make big bluffs as often as GTO will require you to do later on.

68. NFL and College Football Teams that Score a Touchdown with Little Time Left Should Go for Two When Down by Eight. Even if their kicker is 100%, it’s still the right play if they are 40% to succeed with the two-point conversion attempt. You assume you will score another touchdown (and that you are even money in overtime) and go for one the second time if and only if the two-point try worked. I’ll show the calculation in the lengthier explanation, but you should try to work it out for yourself.

73. Lay Me 11-10 on Ten Football Games and You Can Beat Me for 100 While I Cap Your Loss at 55. If your picks are coin flips my expected profit is $2.50 if you bet five games and 5 if you bet ten games with no forgiveness. But capping your losses at 55 (rather than the normal 110) still results in an EV of about 4 for me. In other words if you planned to bet five games with me, I substantially increase my “earn” if I let you bet five more for the same amount without increasing your maximum loss. Do you see why?

75. Chinese Children Help Calculate 1/2 + 2/4 +3/8 + 4/16 + 5/32 + ……..n/ (2 to the nth). A billion Chinese childless couples are told to have a boy and stop having children once they do. How many total children do they have? Half the couples have one. A quarter have two. An eighth have three etc. Analogous to the math problem. Except this time, we know the answer. Two billion (don’t nitpick). So, the series total “2”. There are lots of similar problems that can be done in similar ways.
76. When Coincidences Probably Aren’t. Two things just might be connected but the experts say that is farfetched. Problem is that most experts are not properly adjusting to how unlikely a parlay event will occur. The Challenger explodes on an exceedingly rare sub-freezing day in southern Florida. The first Covid case is walking distance from a lab. An airliner goes down in Iran during the time they are retaliating to an airstrike. These parlays are even less likely than the connection hypothesis. Thus, the hypothesis becomes the favorite without implying that the experts are incompetent in their field.

79. Casinos Should Revamp Their Baccarat Rebate Policy. Presently they give high rollers a small percentage refund on their losses. Instead, they should give back a much higher percentage but only on losses that exceed a high amount. The principle is the same as #73 above. (And there are many other examples.) I contend that both the customer and the casino would like this better as it saves the casino EV (mathematicians have actually exploited the present rebate technique for big money) while doing a better job of ameliorating the pain of those who are feeling it.
80. Two Ideas for Slot Machines. I am withholding most of my ideas for casinos in case one of them wants to pay me what I’m worth to consult for them. But two simple examples would be: 1. Those triple hand draw poker machines pay a big jackpot for three Royals at once. But they could pay an even bigger one for exactly two out of three. Do you see why? 2. Slots and poker machines should add payoffs based on what is obtained over the course of several outcomes in a row. Three pat hands in a row, Ten bars in five pulls. Things like that.