Page 480 - Discrete Mathematics and Its Applications

7.2 Probability Theory 459

EXAMPLE 8 A coin is biased so that the probability of heads is 2/3. What is the probability that exactly four
heads come up when the coin is flipped seven times, assuming that the flips are independent?

7
Solution: There are 2 = 128 possible outcomes when a coin is flipped seven times. The number
ofwaysfourofthesevenflipscanbeheadsisC(7, 4).Becausethesevenflipsareindependent,the
3
4
probability of each of these outcomes (four heads and three tails) is (2/3) (1/3) . Consequently,
the probability that exactly four heads appear is
35 · 16 560
4 3 ▲
C(7, 4)(2/3) (1/3) = = .
3 7 2187
Following the same reasoning as was used in Example 8, we can find the probability of k
successes in n independent Bernoulli trials.

THEOREM 2 The probability of exactly k successes in n independent Bernoulli trials, with probability of
success p and probability of failure q = 1 − p,is
k n−k
C(n, k)p q .

Proof: When n Bernoulli trials are carried out, the outcome is an n-tuple (t 1 ,t 2 ,..., t n ),
where t i = S (for success) or t i = F (for failure) for i = 1, 2,...,n. Because the n trials are
independent, the probability of each outcome of n trials consisting of k successes and n − k
k n−k
failures (in any order) is p q . Because there are C(n, k) n-tuples of S’s and F’s that contain
exactly kS’s, the probability of exactly k successes is

k n−k
C(n, k)p q .
We denote by b(k; n, p) the probability of k successes in n independent Bernoulli tri-
als with probability of success p and probability of failure q = 1 − p. Considered as a
function of k, we call this function the binomial distribution. Theorem 2 tells us that
k n−k
b(k; n, p) = C(n, k)p q .
EXAMPLE 9 Suppose that the probability that a 0 bit is generated is 0.9, that the probability that a 1 bit is
generated is 0.1, and that bits are generated independently. What is the probability that exactly
eight 0 bits are generated when 10 bits are generated?

Solution: By Theorem 2, the probability that exactly eight 0 bits are generated is

8 2 ▲
b(8; 10, 0.9) = C(10, 8)(0.9) (0.1) = 0.1937102445.

JAMES BERNOULLI (1654–1705) James Bernoulli (also known as Jacob I), was born in Basel, Switzerland.
He is one of the eight prominent mathematicians in the Bernoulli family (see Section 10.1 for the Bernoulli
family tree of mathematicians). Following his father’s wish, James studied theology and entered the ministry. But
contrary to the desires of his parents, he also studied mathematics and astronomy. He traveled throughout Europe
from 1676 to 1682, learning about the latest discoveries in mathematics and the sciences. Upon returning to Basel
in 1682, he founded a school for mathematics and the sciences. He was appointed professor of mathematics at
the University of Basel in 1687, remaining in this position for the rest of his life.
James Bernoulli is best known for the work Ars Conjectandi, published eight years after his death. In
this work, he described the known results in probability theory and in enumeration, often providing alternative
proofs of known results. This work also includes the application of probability theory to games of chance and his introduction of the
theorem known as the law of large numbers. This law states that if > 0, as n becomes arbitrarily large the probability approaches 1
that the fraction of times an event E occurs during n trials is within of p(E).