The Language of Argument

2 4 8

C HaP Te r 1 1 ■ C h a n c e s

Rule 2G: Conjunction in General. Given two events, the probability of

their both occurring is the probability of the first occurring times

the probability of the second occurring, given that the first has

occurred. Symbolically:

Pr(h 1 & h 2 ) 5 Pr(h 1 ) 3 Pr(h 2 |h 1 )

Notice that, in the event that h 1 and h 2 are independent, the probability of h 2 is not

related to the occurrence of h 1 , so the probability of h 2 on h 1 is simply the prob-

ability of h 2. Thus, Rule 2 is simply a special case of the more general Rule 2G.

We can extend these rules to cover more than two events. For example,

with Rule 2, regardless of the number of events we might consider, provided

that they are independent of each other, the probability of all of them occur-

ring is the product of each one of them occurring. For example, the chances

of flipping a coin and having it come up heads is one chance in two. What

are the chances of flipping a coin eight times and having it come up heads

every time? The answer is:

1/2 3 1/2 3 1/2 3 1/2 3 1/2 3 1/2 3 1/2 3 1/2

which equals 1 chance in 256.

Probabilities of Disjunctions

Our next rule allows us to answer questions of the following kind: What

are the chances of either an eight or a two coming up on a single throw of

the dice? Going back to the chart, we saw that we could answer this ques-

tion by counting the number of ways in which a two can come up (which is

one) and adding this to the number of ways in which an eight can come up

(which is five). We could then conclude that the chances of one or the other

of them coming up are six in thirty-six, or 1/6. The principle involved in this

calculation can be stated as follows:

Rule 3: Disjunction with Exclusivity. The probability that at least one

of two mutually exclusive events will occur is the sum of the

probabilities that each of them will occur. Symbolically (where h 1

and h 2 are mutually exclusive):

Pr(h 1 or h 2 ) 5 Pr(h 1 ) 1 Pr(h 2 )

To say that events are mutually exclusive means that they cannot both occur

together. You cannot, for example, get both a two and an eight on the same

cast of two dice. You might, however, throw neither a two nor an eight, since

you might throw some other number.

When events are not mutually exclusive, the rule for calculating disjunctive

probabilities becomes more complicated. Suppose, for example, that exactly

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