Gambling Wisdom: More Odds, Risk & Win, Expectations, and Prices
In this series I discuss topics that are well known within the gambling community, but that may be new to players who come to the game of poker from a non-gambling background.
Previously I’ve written on the topic of odds. I want to expand on some of the concepts presented there and introduce the important concepts “risk & win”, “expectation” and “price”. As a brief refresher, we’ve defined odds as the ratio of the number of times some event will give one outcome to the number of times it will give another. A logical extension of this concept occurs when you consider an event that has more than two possible outcomes. Consider in poker the situation where there are cards remaining in the deck that cause you to win the pot, lose the pot, and split the pot. In such situations, it’s useful to consider the ratio of the number of times each outcome will occur. So if there’s 41 cards remaining that cause you to win, 4 that cause you to lose, and 1 that causes a split pot, thinking of odds of 4:1:41 (in my wacky notation of always putting bad outcomes first) is useful. Gambling jargon doesn’t really have a way to say this aloud, because it was developed for wagers with only two outcomes, but nonetheless the concept is important.
Let’s talk about the concept of “risking” and “winning” money in gambling. In the previous odds article, I described comparing the payout for a horse racing wager to the odds of your horse winning. In that example, you believe it’s 2:1 against your horse winning, and you risk 3 to win 7. This “risk X to win Y” notation is somewhat misleading because it doesn’t exactly correspond to the way money would change hands at the betting window. For this wager, if you made a $3 bet you would give $3 to the window, and if you won you would get $10 back - your initial $3 plus $7 that you won. While you may think this is cumbersome initially, it’s the right way to think about bets. Counting the money you initially bet as part of your win isn’t correct. If your horse comes in, your wallet is only $7 thicker than it was before the wager, not $10. In poker, the thinking is even more complex - money from your previous bets at the pot DOES count as part of your win when figuring odds, but money from the current bet does not. This is because previous bets are sunk costs - the money is no longer yours and as such is a real win if you win the pot. In contrast, money from the current bet is not a sunk cost (ie. you still have the option of not betting), and thus does not count as part of your win.
Now I want to introduce the single most important concept for a winning gambler: expectation. We’ve already looked at this topic, although not by name, in the example from the previous paragraph. The decision on whether to bet on the horse is made by comparing the payout (7:3) to the odds against winning (2:1) and noting that the payout is a larger ratio, and hence we should bet. What do I mean by “should” bet? I mean that, if we made a very long string of copies of this bet, with the random outcome re-decided each time, wagering the same amount each time, we would end up with more money at the end of the string of bets than we had at the beginning. In other words, on average, we come out ahead on the bet. In gambling terms, we have a positive expectation on the bet. This is somewhat of an odd construct, since we’re not going to be able to make multiple independently resolved copies of the bet. We’ll only bet the race once. The thinking however, is that if we make a string of different bets of similar size, each with positive expectation, we can expect to make money in exactly the same way as if we cloned the same bet over and over again. Note that the gambling jargon differs here from normal English useage - you lose this bet 2/3 of the time, so a non-gambler might say that they expect to lose the bet. However, that is not taking into account the attractive payoff - expectation in a gambling sense is simply the merger of likelihood, price, and payoff.
Determining if your expectation is positive is fairly easy when you have only two outcomes - you simply compare the odds against winning to the payoff (with the amount you win first and the amount you risk second) and if the payout is the bigger ratio, you bet. As simple and convenient as it is, this computation has some problems. First, it is somewhat awkward in that the odds and payoffs need to be in opposite order for it to work. Second, the comparison method is insufficient in cases where there are multiple outcomes each offering a different payout. As such, I suggest the following more general method:
- List odds from worst outcome to best outcome.
- List risks and wins in the same order, with risks negative
- Multiply the two ratios together, element by element
- Sum the elements together - if the result is positive, you have a positive expectation, if it’s negative you have a negative expectation.
For a simple example, consider a fictional dice game where a single (fair) six sided die is rolled. If it comes up 1,2,3 you lose $4. If it comes up 4,5,6 you win what shows on the face of the die. The process of determining whether you expectation is positive goes as follows:
- Odds of various outcomes: 3:1:1:1
- Outcomes: -4:4:5:6
- Product: -12:4:5:6
- Sum: 3, which is positive, so this game has a positive expectation
Now that you know this game is positive expectation, the question is how good is it? Well, since the sum is a sum across the outcomes, you’ll win that amount on average in a number of trials equal to the number of outcomes. So your expectation is to win $3 in 6 trials, or $.50/trial.
This brings us to the last topic - prices. Most gambling opportunities involve putting down money upfront for a potential win down the road. While expectation thinking simply factors the money put in up front as part of the eventual win or loss, gambling jargon often talks about this scenario in a different way. Specifically, the money that you put down up front (which is your maximum risk) is “buying” a potential win down the road. If the expectation for that win, now NOT factoring in the upfront payment since it’s a sunk cost, is greater than the payment, then you’re “getting a good price” and should play. If the expected payout is too small, you’re “getting a bad price” and shouldn’t play. This way of thinking about odds turns out to be exactly the same as straight expectations thinking but does the same math in a slightly different way. I’ll try to stick with straight expectations notation in my writing, but you’ll see prices crop of from time to time here and elsewhere. It’s just two different ways of saying the same thing.
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