Chances Are: Part III

In my last column, we saw how chances are expressed.

Put 100 marbles in a box, 60 black and 40 white. Reach in and pull out a marble. There is just enough room for your hand, so you can’t see what you grab. You have 40 chances in (or out of) 100 of pulling a white marble. That figure is usually expressed as 4 out of 10, or 2 out of 5. As a fraction, it becomes a 2/5 chance; as a decimal, a 0.4 chance; as a percent, a 40 percent chance.

How about the odds? There are 60 black and 40 white, or 60-to-40 against getting a white marble. That can be expressed more simply as 6-to-4, or odds of 3-to-2 against. You often see odds expressed in such a form that the two numbers add up to 100, or 10. Thus, it would be equally likely to see the preceding expressed in any of the given ways. You can see from this where the expression “50-50,” meaning an equal likelihood of either one of two events, came from.

The two figures that make up the odds are often written separated by a colon or, sometimes, less often, by a hyphen. So, odds of 60-to-40 are often expressed as 60:40, or sometimes as 60-40. When read aloud, both are pronounced as if they were written “60-to-40,” and all three mean the same.

The odds are not always against you. In the previous example, you have a 60 percent chance of drawing a black marble. The odds are 60-to-40 in your favor of that event.

This, too, could be expressed variously as 6-to-4, 3-to-2, 60:40, 6:4, 3:2, or, less commonly, 60-40, 6-4, 3-2 in favor.

Odds are often expressed such that the lower figure of the two becomes a 1. If your marble box held 19 marbles, 2 of which were white and the rest black, it would be 17-to-2 against pulling a white marble. This is often expressed as 8½-to-1, or 8.5-to-1.
To figure what the odds to 1 are, divide the larger number by the smaller. For example, in the preceding, we divided 17 by 2 to get 8½. (We also divided 2 by 2 to get 1.) This method is the same as reducing a fraction. Divide both numerator and denominator by the same number. This is also how we turned odds of 60-to-40 to 6-to-4. We divided both numbers by the same number, in that case, 10. And to get odds of 3-to-2, we divided both numbers by 20. Odds of 3-to-2 might be further reduced if you wanted to use 1 for the lower number. That would be 1½ to 1.

You can go the reverse direction if you want to express odds as chances. To change odds of 3-to-2 into chances, add the two numbers together, and divide that total by the lower number. Adding, you get 5, and dividing produces 2/5, or, as a decimal, 0.4. That is also 40 percent, and, in the other methods of expressing it, chances of 2 in 5, or 2 out of 5.

So when the weatherman says, “The is a 30 percent chance of rain today,” you know that he means that whenever the identical weather patterns come up, 30 times out of 100 it rains, and (by subtracting 30 from 100) 70 times it doesn’t. You know also that the odds are 7-to-3 against rain. If a friend offers to bet you $10, and gives you 2-to-1 odds, that it will rain, don’t take him up on it, because the odds against rain are worse than that. That is, in 100 similar situations, it rains 30 times. He gives you $20 each time it rains, or $600 altogether. You give him $10 each of the 70 times it doesn’t rain, or $700. You’re out $100, or an average of $1 per bet. (You lose $100 over 100 bets. Divide $100 by 100 to get the per-bet average.) On the other hand, if he offers 3-to-1, that’s a good bet for you. You get $30 times 30, or $900, and lose $10 70 times, or $700, for a net profit of $200, or $2 per bet. (My own opinion of weather prognosticators making such a statement is that they must not know much about their field of expertise, because they’re wrong 70 percent of the time.)

All this has relevance to poker because you often see figures about how likely it is to make a hand, what a pot is offering compared to what it costs you to get in (pot odds), and so on. For example, in no-limit hold’em, with one card to come, if you have the A K and the board is 10 9 6 3, you have 9 chances in 46 of making your flush. That is, you have a little less than a 1/5 chance of spiking a spade. Also, the odds are 37-to-9 against your making the hand, or a little worse than 4-to-1 against you. If you faced an all-in bet of $100 and the pot already contained $400, pot odds would be 5-to-1 and calling would be the right decision. If the pot contained only $250, calling would not be correct.

Now that you can speak knowledgeably and correctly about odds and chances, you can bet more intelligently. If you do that, as Johnny Mathis sang more than 50 years ago, “The chances are your chances are awfully good.” ♠

Michael Wiesenberg has been a columnist for Card Player since 1988. He has written or edited many books about poker, and has also written extensively about computers. His crossword puzzles are syndicated in newspapers and appear online. Send orchids, opprobrium, and offerings to [email protected].