Practical Probability — Part VIII

All gamblers must understand the concept of equity. Equity should be the guiding principle for almost every decision you will ever make. Equity is often referred to as expectation, mathematical expectation, or expected value. The simplest way to conceptualize equity is to think of it as the average value (a positive or negative numerical value) of any given situation. The reason that I prefer to use the term equity is that the use of an expression like “expected value” sometimes causes people to think that it’s the most likely outcome. On a few occasions, it is both the average value and the most likely value. For example, if you roll a pair of dice, the expected value of their sum is 7, which is also the most likely outcome. If you roll one die, the expected value is 3.5, but 3.5 can never actually occur.

If there is a situation with a variety of possible outcomes and each outcome has some probability of occurring, a not so rigorous definition of equity is the sum of the value of each outcome multiplied by the probability of that outcome. Here are some simple examples of figuring equity. You have just made two spade flushes. Someone says that it seems that spades are more likely to come out than any other suit. You ask if he wants to bet, and he agrees to do so. You say, “If the first card dealt is a spade, I’ll give you $5,000, and if it isn’t, you give me $5,000.” He replies, “If that first card dealt is a spade, I get the $5,000, but if it isn’t, you get only $2,000.”

1. What is the equity of the bet you proposed?

2. What is the equity of the bet he proposed?

3. Should you accept his offer?

You should remember that the sum of the probabilities of all possible outcomes must equal 1. In this case, the probability that the first card is a spade is 0.25 and the probability that it’s a non-spade is 0.75. So, the answer to question No. 1 is that your equity is equal to 0.25 times -$5,000 plus 0.75 times +$5,000, which equals -$1,250 plus $3,750. Your equity from the bet you proposed is +$2,500. In reality, you will either win or lose $5,000 and never win exactly $2,500, but $2,500 is what the bet is worth to you — your equity.

His proposal is clearly worse. Now your expectation can be calculated as 0.25 times -$5,000 plus 0.75 times +$2,000, which equals -$1,250 plus $1,500. Your equity from making this new bet is only +$250.

Since $250 is positive (greater than zero), you should still take the bet. In poker, you will sometimes have a choice of two actions, both of which have a positive expectation, and you must choose the one with the greatest value. For example, you make the nuts on the river. The pot is $1,500 and you and your opponent both have $2,000 left. You estimate that a bet of $800 will be called 70 percent of the time, a bet of $1,200 will be called 50 percent of the time, and a bet of $2,000 will be called 40 percent of the time.

1. Does the amount of money already in the pot make any difference?

2. What is your equity for each bet amount?

3. What is the proper amount to bet?

In this situation, the amount of money that’s already in the pot doesn’t matter. You will win it no matter what happens. Your additional equity consists of the amount that you bet times the frequency of his call. Therefore, your bets are worth $560, $600, and $800, respectively. You should bet the full $2,000. Your total equity is the pot of $1,500 that you will definitely win plus the $800 that you will win, on average, from the river bet, or $2,300.

Now let’s suppose in this same situation that you have nothing, and are thinking about bluffing. Does the amount of money that’s already in the pot make any difference? Should you bluff, and if so, which of the three bet sizes should you choose? In this case, the money in the pot is crucial, because you can win it only when your opponent folds. In the first case, you would always win it. The amounts you won from his calls in the first case are now the amounts you will lose when he calls. Thus, your bet of $800 will be called 70 percent of the time, costing you $560, but you will win the pot of $1,500 30 percent of the time, earning $450. The $800 bluff has equity of -$110 — no good. The bluff of $1,200 will be called 50 percent of the time, costing you $600, but you will win the pot 50 percent of the time, earning $750. This bluff has equity of +$150. That’s pretty good, but could the big bluff be even better? Well, 40 percent of the time, you will lose $2,000, costing yourself $800, but 60 percent of the time, you will win $1,500, earning $900. The big bluff has equity of +$100. That is good, but the $1,200 bluff has the largest equity, so you should make that bluff.

Note that you gain equity by betting, both with the nuts and as a bluff, but the optimum bet size is different in each case. The situation described here is typical of many situations in no-limit hold’em, in which it pays to be aggressive.

In the first column of this series, I said that odds are often the easiest way for a gambler to apply probability to making betting decisions. Odds compare the outcomes that aren’t in the specified group to those in it. Thus, there are 39 non-spades. If we divide this by 13 spades, we get 3-1.

I thought it would be nice to recommend an easily accessible text of probability theory and practice. I picked up Probability for Dummies by Deborah Rumsey, Ph.D., published by Wiley, considered to be a respected publisher of mathematical and scientific texts. I had barely begun to review the book when I saw on the first page of Chapter 1 (Page 9) the following: “The term odds, however, isn’t exactly the same as probability. Odds refers to the ratio of the denominator of a probability to the numerator of a probability. For example, if the probability of a horse winning a race is 50 percent (1/2), the odds of this horse winning are 2 to 1.” I stared at that statement in stunned disbelief. Obviously, the correct odds are even money, or 1-to-1. By her formula, ½ divided by ½ = 1-to-1. She’s a mathematics professor. How could she get it wrong? What about editors and proofreaders? (We Card Player writers are lucky to have Editor in Chief Steve Radulovich and his great editorial team to catch such gaffes.) I haven’t completed the book, but the rest of it seems like a reasonable, although somewhat sloppy, introduction to probability theory.

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.