25

51

26

51

pqp→q

TTT

TFF

FTT

FFT

90 conditional convergence

The

TRUTH TABLE

of the conditional is motivated

by intuition. Consider, for example, the statement:

If Peter watches a horror movie, then he eats

popcorn.

The statement is certainly true if we observe Peter

watching a horror movie and eating popcorn at the

same time (that is, if the antecedent and consequent are

both true), but false if antecedent is true but the conse-

quent is false (that is, Peter is watching a horror movie

but not eating popcorn). This justifies the first two

lines of the truth table below.

The final two lines are a matter of convention. If

Peter is not watching a horror movie, that is, if the

antecedent is false, then the conditional statement as a

whole is moot.

FORMAL LOGIC

, however, requires us to

assign a truth-value to every statement. As watching a

romance movie and eating (or not eating) popcorn does

not imply that the conditional statement is a lie, we go

ahead and assign a truth-value “true” to the final two

lines of the table:

This convention does lead to difficulties, however. Con-

sider, for example, the following statement:

If this entire sentence is true, then the moon is

made of cheese.

Here the antecedent pis the statement: “the entire sen-

tence above is true.” The consequent is: “the moon is

made of cheese.” Notice that pis true or false depend-

ing on whether the entire statement p→qis true or

false. There is only one line in the truth table for which

pand p→qhave the same truth-value, namely the first

one. It must be the case, then, that p, q, and p→qare

each true. In particular, qis true. Logically, then, the

moon must indeed be made of cheese.

See also

ARGUMENT

;

BICONDITIONAL

;

CONDITION

—

NECESSARY AND SUFFICIENT

;

SELF

-

REFERENCE

.

conditional convergence See

ABSOLUTE CONVER

-

GENCE

.

conditional probability The probability of an

EVENT

occurring given the knowledge that another event has

already occurred is called conditional probability.

For example, suppose two cards are drawn from

a deck and we wish to determine the likelihood that

the second card drawn is red. Knowledge of the first

card’s color will affect our probability calculations.

Precisely:

i. If the first card is black, then the probability that the

second is red is 26/51 (there are 26 red cards among

the remaining 51 cards),

ii. If the first card is red, then the probability that the

second is also this color is now only 25/51.

(If we have no knowledge of the color of the first card,

then the chances that the second card is red are 1/2.)

If Aand Bare two events, then the notation A|Bis

used to denote the event: “Aoccurs given that event B

has already occurred.” The notation P(A|B) denotes the

probability of Aoccurring among just those experi-

ments in which Bhas already happened. For instance,

in the example above:

P(the second card is red | the first card is black) =

P(the second card is red | the first card is black) =

If, in many runs of an experiment, event Boccurs b

times, and events Aand Boccur simultaneously a

times, then the proportion of times event Aoccurred

when Bhappened is a/b. This motivates the mathemati-

cal formula for conditional probability:

As an example, suppose we are told that a card drawn

from a deck is red. To determine the probability that

that card is also an ace we observe: