B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

398 FUNDAMENTALS OF PROBABILITY THEORY
A sum of seven can be obtained by the six combinations: (1, 6), (2, 5), (3, 4), (4, 3)
(5, 2), and (6, 1). Hence, the event "a seven is thrown" is the union of six outcomes, each
with probability 1/36. Therefore,
1 1 1 1 1 1 1
36 36 36 36 36 36 6
P("a seven is thrown") = - + - + + - + - + - = -
Example 8 .4
A coin is tossed four times in succession. Determine the probability of obtaining exactly two
heads.
I
I
;il'l
A total of 2 4 = 16 distinct outcomes are possible, all of which are equally likely because
of the symmetry of the situation. Hence, the sample space consists of 16 points, each with
probability 1/16. The 16 outcomes are as follows:
HHHH
HHHT
HHTH
---+HHTT
HTHH
---+HTHT
---+ HTTH
HITT
IT1T
TITH
TTHT
---+TTHH
THTT
---+THTH
---+THHT
THHH
Six out of these 16 outcomes lead to the event "obtaining exact! y two heads" (arrows).
Because all of the six outcomes are disjoint (mutually exclusive),
. . 6 3
P( obtamrng exactly two heads) = - = -
16 8
In Example 8.4, the method of listing all possible outcomes quickly becomes unwieldy
as the number of tosses increases. For example, if a coin is tossed just 10 times, the total
number of outcomes is 1024. A more convenient approach would be to apply the results of
combinatorial analysis used in Bernoulli trials, to be discussed shortly.
Conditional Probability and Independent Events
Conditional Probability: It often happens that the probability of one event is influenced
by the outcome of another event. As an example, consider drawing two cards in succession
from a deck. Let A denote the event that the first card drawn is an ace. We do not replace the card
drawn in the first trial. Let B denote the event that the second card drawn is an ace. It is evident
that the probability of drawing an ace in the second trial will be influenced by the outcome of
the first draw. If the first draw does not result in an ace, then the probability of obtaining an ace
in the second trial is 4/5 1. The probability of event B thus depends on whether event A occurs.
We now introduce the conditional probability P(BIA) to denote the probability of event B
when it is known that event A has occurred. P(B IA) is read as "probability of B given A."