outcomes of one action do not affect the outcomes of
the actions that follow.
To illustrate, imagine tossing a coin and then casting
a die. Each outcome of the coin toss—heads or tails—
could be accompanied by any one of the six possible
outcomes from casting the die: the numbers 1 through 6.
The probability-tree diagram (shown above) is used to
make this explicit. In particular, it shows that 12 differ-
ent results are possible from performing these two
actions. Computing the probability of any desired set of
outcomes is now straightforward. For example, the
chances of tossing a head together with an even number
are = . Similarly, the probability of tossing a tail
together with a 5 or a 6 is = .
This model illustrates the additive property in
probability theory. In our example:
P(heads and an even number or tails and a 5 or a 6)
=
= +
= P(heads and an even number) + P(tails and a 5 or a 6)
In general, if Arepresents one set of desired outcomes
and Banother set of outcomes having none in common
with A, then
P(Aor B) = P(A) + P(B)
In particular, the two events “A” and “not A” are dis-
joint, that is, have no outcomes in common. As we are
certain that Aeither will or will not occur, P(Aor not
A) = 1, we have, by this rule:
P(not A) = 1 – P(A)
For example, the probability of not rolling a two when
casting a die is P(not 2) = 1 – p(rolling 2) = 1 –= .
If two events Aand Bdo have outcomes in com-
mon, then the above rule is modified to read:
P(Aor B) = P(A) + P(B) – P(Aand B).
The term P(Aand B) is subtracted to counter the dou-
ble count of outcomes common to A and B.
2. The Square Model and the Multiplication Rule
The square model for probability theory uses a square
to represent the set of results in performing an experi-
ment a large number of times. For example, in tossing a
coin, we would expect, on average, half the outcomes
to be heads and half to be tails. We represent this by
dividing the square into two portions of equal area.
The left portion now represents a set of experiments in
its own right. In next casting a die, we would expect,
on average, half of these outcomes to yield an even
number and half to yield an odd number. This divides
the heads region into two equal subportions. The right
portion of the square also represents a set of experi-
ments in its own right. In casting a die we would
expect one-sixth of the outcomes to be a 1, one-sixth to
be 2, and so on, all the way through to one-sixth of the
outcomes being 6. This divides the tails region into six
portions of equal area.
Now it is easy to read off probabilities regarding
combinations of outcomes. For example, the outcome
of “heads followed by an even number” is represented
by one half of half the square, that is:
P(heads and an even number) = ×=
The outcome “tails and a 5 or a 6” is represented by
one sixth of half the square plus another sixth of half
the square. Thus:
P(tails and 5 or 6) = ×+ ×=
This model illustrates the multiplicative property in
probability theory. In our example:
1
–
6
1
–
2
1
–
6
1
–
2
1
–
6
1
–
4
1
–
2
1
–
2
5
–
6
1
–
6
2
–
12
3
–
12
3 + 2
–––
–
12
1
–
6
2
–
12
1
–
4
3
–
12
416 probability
Probability models