Page 42 - Probability, Random Variables and Random Processes

PROBABILITY [CHAP 1

of a failure on the ith trial. Then the probability of no successes is, by independence,
P(A, n A, n - . . n A,) = P(Al)P(A2) . - . P(A,) = (1 - p)" (1.87)
Hence, the probability that at least 1 success occurs in the first n trials is 1 - (1 - p)".
(b) In any particular sequence of the first n outcomes, if k successes occur, where k = 0, 1, 2, . . . , n, then
n - k failures occur. There are (9
such sequences, and each one of these has probability pk(l - P)"-~.
Thus, the probability that exactly k successes occur in the first n trials is given by - pyk.

(c) Since Ai denotes the event of a success on the ith trial, the probability that all trials resulted in
successes in the first n trials is, by independence,
P(Al n A, n . + n An) = P(A,)P(A,) . . P(A,,) = pn (1.88)
Hence, using the continuity theorem of probability (1.74) (Prob. 1.28), the probability that all trials
result in successes is given by
0 p<l
(1-1 ) m i 1 ) n i ) n-cc {l p= 1
P OXi =P lim r)Ai = limp nXi = limpn=

Let S be the sample space of an experiment and S = {A, B, C), where P(A) = p, P(B) = q, and
P(C) = r. The experiment is repeated infinitely, and it is assumed that the successive experiments
are independent. Find the probability of the event that A occurs before B.
Suppose that A occurs for the first time at the nth trial of the experiment. If A is to have occurred
before B, then C must have occurred on the first (n - 1) trials. Let D be the event that A occurs before B.
Then

where D, is the event that C occurs on the first (n - 1) trials and A occurs on the nth trial. Since Dm's are
mutually exclusive, we have

Since the trials are independent, we have

Thus,

1.63. In a gambling game, craps, a pair of dice is rolled and the outcome of the experiment is the sum
of the dice. The player wins on the first roll if the sum is 7 or 11 and loses if the sum is 2,3, or 12.
If the sum is 4, 5, 6, 8, 9, or 10, that number is called the player's "point." Once the point is
established, the rule is: If the player rolls a 7 before the point, the player loses; but if the point is
rolled before a 7, the player wins. Compute the probability of winning in the game of craps.

Let A, B, and C be the events that the player wins, the player wins on the first roll, and the player gains
point, respectively. Then P(A) = P(B) + P(C). Now from Fig. 1-3 (Prob. IS),