The Language of Argument

2 4 4

C HaP Te r 1 1 ■ C h a n c e s

We can make our probability claims more precise by using numbers.

Sometimes we use percentages. For example, the weather bureau might

say that there is a 75 percent chance of snow tomorrow. This can naturally

be changed to a fraction: The probability is 3/4 that it will snow tomorrow.

Finally, this fraction can be changed to a decimal expression: There is a 0.75

probability that it will snow tomorrow.

The probability scale has two end points: the absolute certainty that the

event will occur and the absolute certainty that it will not occur. Because you

cannot do better than absolute certainty, a probability can neither rise above

100 percent nor drop below 0 percent (neither above 1, nor below 0). (This

should sound fairly obvious, but it is possible to become confused when

combining percentages and fractions, as when Yogi Berra was supposed

to have said that success is one-third talent and 75 percent hard work.) Of

course, what we normally call probability claims usually fall between these

two end points. For this reason, it sounds somewhat peculiar to say that

there is a 100 percent chance of rain and just plain weird to say the chance of

rain is 1 out of 1. Even so, these peculiar ways of speaking cause no proce-

dural difficulties and rarely come up in practice.

A Priori Probability

When people make probability claims, we have a right to ask why they

assign the probability they do. In Chapter 8, we saw how statistical proce-

dures can be used for establishing probability claims. Here we will examine

the so-called a priori approach to probabilities. A simple example will bring

out the differences between these two approaches. We might wonder what

the probability is of drawing an ace from a standard deck of fifty-two cards.

Using the procedure discussed in Chapter 8, we could make a great many

random draws from the deck (replacing the card each time) and then form a

statistical generalization concerning the results. We would discover that an

ace tends to come up roughly one-thirteenth of the time. From this we could

draw the conclusion that the chance of drawing an ace is one in thirteen.

But we do not have to go to all this trouble. We can assume that each of the

fifty-two cards has an equal chance of being selected. Given this assumption,

an obvious line of reasoning runs as follows: There are four aces in a stand-

ard fifty-two-card deck, so the probability of selecting one randomly is four

in fifty-two. That reduces to one chance in thirteen. Here the set of favorable

outcomes is a subset of the total number of equally likely outcomes, and to

compute the probability that the favorable outcome will occur, we merely

divide the number of favorable outcomes by the total number of possible

outcomes. This fraction gives us the probability that the event will occur on

a random draw. Since all outcomes here are equally likely,

Probability of drawing an ace 5

number of aces

total number of cards

5

4

52

5

1

13

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