Practical Probability — Part IX

“We must believe in luck! For how else could we explain the successes of those we don’t like?”

— Jean Cocteau

Each of the following problems consists of two possible outcomes. Your mission, if you choose to accept it, is to decide if one is more likely than the other, and if so, by how much. If you have difficulty in figuring out exactly how much more likely one outcome is than the other, try to approximate the answer, or even guess at the answer.

1. Two dice are rolled and their sum is (a) 7 or (b) 8.
2. Three dice are rolled and their sum is (a) 9 or (b) 10.
3. You are dealt two aces; (a) they are both the same color or (b) they are mixed colors.
4. You are dealt five cards; they are (a) A K Q J 10 or (b) 8 3 K 7 10.

“People always call it luck when you’ve acted more sensibly than they have,” said Anne Tyler. Never make the mistake of thinking that luck doesn’t exist. It does. The unfortunate thing is that it is impossible to predict in advance when you will be lucky or unlucky. The best way to handle decision-making is by focusing on taking actions that will maximize your equity. Answer to No. 1: If you bet that two dice will show a sum of 7 before a sum of 8, you will average winning one bet for every 36 rolls. One easy way to solve this problem is by counting possibilities. In 36 rolls of the dice, 7 appears six times (4-3, 3-4, 5-2, 2-5, 6-1, and 1-6), while 8 appears five times (5-3, 3-5, 6-2, 2-6, and 4-4). That is the reason that the fair odds are 6-to-5. The reason that 4-4 occurs only once while the other combinations occur twice can be easier to conceptualize if you think of rolling one die twice. For example, two fours occur only when the first roll is a 4 and the second roll is a 4, while a 5 and a 2 can occur when the first roll is a 5 and the second is a 2, or when the first is a 2 and the second is a 5.

“The winds and the waves are always on the side of the ablest navigators,” said Edward Gibbon. While I agree with the idea that the best players will probably get the best results in the long run, I certainly don’t think that the “poker gods” are always on their side. Perhaps a better quote is this one: “What helps luck is a habit of watching for opportunities, of having a patient, but restless mind, of sacrificing one’s ease or vanity, of uniting a love of detail to foresight, and of passing through hard times bravely and cheerfully,” said Charles Victor Cherbuliez. In any case, it is clear that everyone will be lucky and unlucky. Those who will do best are those who possess the most skills and who are prepared psychologically to deal with whatever happens. Answer to No. 2: The sum of 10 is more likely than the sum of 9. It is a 27-to-25 favorite. As in the previous problem, the most accurate answer is found by the laborious process of calculating all of the possible ways that each outcome can occur. This is a good spot to use a combination of estimation and intuition to get an approximate answer. Rolls that have a sum of 9 include one triple, 3-3-3, while those that have a sum of 10 don’t include any. Since a triple is less likely than a normal number (by extension of the 4-4 argument above), 9 is less likely than 10.

Another way you might attempt to approximate the answer is by thinking that the average roll of one die is 3.5. Therefore, the average roll for three dice is 10.5, and since 10 is closer to 10.5 than 9 is, it is probably more likely. For example, if the average man is 69 inches tall, it is logical to expect that a height of 69 inches will occur more often than one of 65 inches. It is generally true that for typical distributions, values closer to the average are more likely. This method won’t always work, as it is easy to find situations in which this isn’t the case. For example, the height of the average human might be 66 inches. However, the height of 69 inches, the average man, or 63 inches, the average woman, might occur more often than the human average of 66 inches.

“Probability is the very guide of life,” said Cicero. This bit of ancient wisdom from a Roman philosopher is actually more profound than it might appear. At first you think that probability is a good guide to certain decisions, especially financial ones, but it doesn’t really help much in making decisions in areas that everyone considers essential, such as those that relate to physical or emotional health. Even in these areas, there are studies that show what is probably good and what isn’t. (Smoking is bad. Exercise is good. Wealth beyond a reasonable level of comfort doesn’t make you happier, but helping others and having pets does.) Answer to No. 3: By now you probably have guessed that the theme of all of the problems is counting up the possibilities for each outcome and comparing them. You can have the same color aces in only two ways (spade-club and heart-diamond), but you can have mixed colors in four ways (heart-spade, heart-club, diamond-spade, and diamond-club). This means that mixed colors are twice as likely as matching colors.

Probability is the one branch of mathematics “in which good writers frequently get results that are entirely erroneous,” said Charles S. Pierce. Probabilistic thinking doesn’t come naturally to most people. Our intuition often leads us astray. I mentioned in my last column a huge gaffe on the first page of text in the popular Probability for Dummies. Since that time, I have started another, much more rigorous book, and it too has errors, although the author tells me that they have been corrected in later editions. Answer to No. 4: They are both equally likely. All groups of five specific cards are equally likely. Instinctively, we assume that a royal flush is less likely than garbage, and it is. However, we also assume that it is less likely than specific garbage, and it isn’t. This same type of error occurs when we look at a series of coin flips. We think HTTHTHH is more likely than TTTTTTT. Each sequence consists of a series of flips with a probability of .5. The probability for the series of seven flips is .5 to the 7th power, or about .0078.

This type of erroneous thinking leads to something known as the gambler’s fallacy. Basically, this is the mistaken belief that a series of deviations in one direction will result in the next few results tending to balance out that abnormal sequence. Have you ever seen a lot of heads in a row and felt that tails was due? Have you ever watched a roulette table where black has come up several times and “known” that it was right to bet on red? If you have, you fell into the trap known as the gambler’s fallacy. Since heads/tails or red/black are random outcomes, their probability is unaffected by what has just occurred. This leads us to a discussion of independence that will have to wait for my next column.

Steve “Zee” Zolotow, aka The Bald Eagle, is a successful games player. He currently devotes most of his time to poker. He can be found at many major tournaments and playing on Full Tilt, as one of its pros. When escaping from poker, he hangs out in his bars on Avenue A — Nice Guy Eddie’s at Houston and Doc Holliday’s at 9th Street — in New York City.