P(heads and an even number)
= half of half the square
= ×
= P(even) ×P(heads)
P(tails and {5, 6})
= one-third of half the square
= P({5,6}) ×P(tails)
In general, if Aand Brepresent two sets of desired
outcomes from two different sets of experiments, then
the probability of obtaining both Aand Bwhen per-
forming the experiments in succession is:
P(Aand B) = P(A) ×P(B)
We are assuming here that the outcomes of one experi-
ment have no effect on the outcomes of the second, that
is, that the two experiments are
INDEPENDENT EVENTS
.
Philosophical Difficulties
Probability, as defined thus far, relies on our ability to
count outcomes. If the set of outcomes is infinitely large,
then the issue of counting is meaningless. Nonetheless we
may still have an intuitive understanding of “likelihood”
in these situations. For example, in spinning a compass
point, we still feel certain that the probability of the
pointer landing between north and east is 1/4, even
though there are infinitely many places for the pointer to
stop in the one desired quarter of the rim of the compass.
If an integer is chosen at random, we suspect that the
likelihood of it being even is 1/2—after all, there are
only two possibilities, even or odd, each representing
half the possible outcomes. Thus determining “proba-
bility” relies on an ability to assign a relative measure
of size to sets of points, or outcomes, even if those sets
might be infinite. This is a difficult issue, and one that
caused much confusion during the 19th and early 20th
century. (See
AREA
and B
ERTRAND
’
S PARADOX
.)
A second difficulty lies in the fact that Cardano’s
definition of probability is circular: the probability of
any outcome is determined by knowing beforehand
which outcomes are equally probable.
How Probability Is Understood Today
In tossing a coin, for example, we are generally willing
to say that just two outcomes are possible—heads or
tails—and we believe that it is appropriate to assign a
probability of 1/2 for each occurring. Folks of a con-
trary disposition may argue, however, that more than
two outcomes could occur (the coin might land on its
side, for example) and that the values of probability
should be assigned differently. Certainly the issues aris-
ing in Bertrand’s paradox, for instance, show that the
notion of randomness is subject to personal understand-
ing. For a meaningful mathematical discussion to take
place, it must therefore be agreed upon beforehand
which outcomes are deemed within the range of possi-
bility, and what the probability of all sets of outcomes
will be. (This takes Cardano’s approach further—not
only must the equally likely outcomes be specified, but
also all probabilities must be declared at the outset.)
Thus, in any discussion of probability theory, a
mathematician today will state at the outset:
1. The sample space S: the set of all outcomes consid-
ered possible
2. A probability measure P: a rule that assigns to any
event ASa number P(A) called the probability
of A.
The probability measure is to satisfy these three rules:
i. For any event A, P(A) is a number between 0 and 1.
ii. P(S) = 1. (That is, in any run of the experiment, an
outcome will occur.)
iii. If two events Aand Bhave no outcomes in com-
mon, then P(Aor B) = P(A) + P(B).
For simple finite models, such as the act of casting
a die, this model encodes the approach developed by
Cardano, Pascal, and Fermat, but it also extends this
thinking to more complex systems. For example, in
throwing a dart at a dartboard, probabilities can be
defined as ratios of areas. The probability of throwing
a bull’s-eye, for instance, is the ratio of the area of the
bull’s-eye to the area of the entire board. This defini-
tion satisfies axioms i, ii, and iii above.
The key then to analyzing any random phe-
nomenon is to appropriately define a probability mea-
sure. Different probability measures can lead to
different results. Physicists and scientists are challenged
then to find the measure that best reflects one’s intuitive
understanding of the phenomenon being discussed.
See also
CONDITIONAL PROBABILITY
;
EVENT
;
HIS
-
TORY OF PROBABILITY AND STATISTICS
(essay); K
RUSKAL
’
S
⊇
1
–
2
1
–
2
probability 417