Chances Are: Part II

In my last column, we saw how writers, particularly those who compose the rules for sweepstakes and lotteries and those who try to explain the math involved to laymen, confuse chances and odds.

Chances are the probability or likelihood of an event, expressed either as one number in (or out of) another (larger) number, a decimal number or fraction less than 1, or a percent.

Odds are also the probability of an event, the ratio of probability that something is more likely to occur than something else. In most cases, odds express the unlikelihood of an event.

The numbers used to express the two are closely related, and therein lies the confusion.

For example, a box contains three marbles, two black and one white, and you shake one marble through a hole in the top just large enough to let one marble out. (This is so that you can’t see the marble you choose.) You note the color of this marble, put it back through the hole, and repeat the experiment. If you did this 300 times, you could be reasonably certain that you would have shaken out a white marble about 100 times and a black marble about 200 times. That is, in looking back on the results, you could say that the chances of getting a white marble were 100 in 300, or 100 out of 300. Since both numbers are exactly divisible by 100, these chances would usually be expressed as 1-in-3, or 1 out of 3. You could be said, then, to have a 1 in 3 chance, or a 1 out of 3 chance, of picking a white marble. Express that as a fraction, and you also could be said to have a one-third chance of picking a white marble. Chances are often expressed as a decimal, so it also would be correct to state that you have a .33333 chance of picking a white marble. Since 1/3 becomes what is called a repeating decimal, the decimal number could be expressed as 0.3, 0.33, 0.333, 0.333333333333333, and so on. As a percent, it becomes a 33 percent chance of picking a white marble. That is expressed precisely as 33-1/3 percent, or, as before, 33.3 percent, 33.333333 percent, and so on.

In your 300 pulls, you also would have had 200 chances out of 300 of picking a black marble, which could be expressed as chances of 2-in-3, 2 out of 3, 2/3, .6667, or 66-2/3 percent. (The decimal could be expressed as 0.7, 0.67, 0.667, 0.6666666666666667, and so on, and the percent as 67 percent, 66.7 percent, 66.67 percent, 66.667 percent, 66.6666666666666667 percent, and so on.)

And before you look at the odds involved in this situation, notice that both chance and chances are used. Usually, the singular is used when directly following the number, and the plural when introducing it. That is, you generally would say, “My chances of winning are one-third,” but, “I have a one-third chance of winning.” You also might say, “I have one chance in three of winning.” Take your pick.

Chances compare the likelihood of a particular event with all events. (One white marble out of three total marbles.) Odds express the likelihood of a particular event compared with the likelihood of another event. In the preceding example, you are twice as likely to get a black marble as a white one, so the odds are 2-to-1 against getting a white marble. Another way of figuring this is noting that there are two black marbles for each white one.

You can see how chances and odds relate here. To express the chances of an event you are interested in, compare it with the total, and either express that as a decimal number, a fraction, or a percent, or express the chances as two numbers, the first in (or out of) the total. To express the odds, take the event you are interested in, subtract the figure associated with it from the total, and then express it as this difference to the first.

Here is a more complicated example:

Put 100 marbles in the box, 60 black and 40 white. 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; and as a percent, a 40 percent chance.

Remember that there are two ways to change a decimal into a percent. One is to think of it as a fraction, and multiply that by 100: 0.4 is expressed as 4/10; multiply by 100, and you get 40; then, append percent, and you get 40 percent. The other way is to move the decimal point two places to the right, and add percent: Think of 0.4 as 0.40, and you see how it becomes 40 percent.
Next time, we’ll see how odds work. ♠

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 cheers, censure, and contributions to [email protected].