Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 7] PROBABILITY 131

Repeated Trials with Two Outcomes, Bernoulli Trials, Binomial Experiment

Now consider an experiment with only two outcomes. Independent repeated trials of such an experiment are
called Bernoulli trials, named after the Swiss mathematician Jacob Bernoulli (1654–1705). The term independent
trials means that the outcome of any trial does not depend on the previous outcomes (such as tossing a coin).
We will call one of the outcomessuccessand the other outcomefailure.
Letpdenote the probability of success in a Bernoulli trial, and soq= 1 −pis the probability of failure.
Abinomial experimentconsists of a fixed number of Bernoulli trials. A binomial experiment withntrials and
probabilitypof success will be denoted by

B(n, p)

Frequently, we are interested in the number of successes in a binomial experiment and not in the order
in which they occur. The next theorem (proved in Problem 7.27) applies. We note that the theorem uses the
following binomial coefficient which is discussed in detail in Chapter 5:

(

n

k

)

=

n(n− 1 )(n− 2 )...(n−k+ 1 )

k(k− 1 )(k− 2 )... 3 · 2 · 1

=

n!

k!(n−k)!

Theorem 7.7: The probability of exactlyksuccesses in a binomial experimentB(n, p)is given by

P(k)=P(ksuccesses)=

(

n

k

)

pkqn−k

The probability of one or more successes is 1−qn.

EXAMPLE 7.12 A fair coin is tossed 6 times; call heads a success. This is a binomial experiment withn= 6
andp=q=^12.

(a) The probability that exactly two heads occurs (i.e.,k=2) is

P( 2 )=

(

6

2

)(

1

2

) 2 (

1

2

) 4

=

15

64

≈ 0. 23

(b) The probability of getting at least four heads (i.e.,k= 4 ,5or6)is

P( 4 )+P( 5 )+P( 6 ) =

(

6

4

)(

1

2

) 4 (

1

2

) 2

+

(

6

4

)(

1

2

) 4 (

1

2

) 2

+

(

6

6

)(

1

2

) 6

=

15

64

+

6

64

+

1

64

=

11

32

≈ 0. 34

(c) The probability of getting no heads (i.e., all failures) isq^6 =

(

1

2

) 6

=

1

64

, so the probability of one or more

heads is 1−qn= 1 −

1

64

=

63

64

≈ 0 .94.

Remark: The functionP(k)fork= 0 , 1 , 2 ,...,n, for a binomial experimentB(n, p), is called thebinomial
distributionsince it corresponds to the successive terms of the binomial expansion:

(q+p)n=qn+

(

n

1

)

qn−^1 p+

(

n

2

)

qn−^2 p^2 +···+pn

The use of the termdistributionwill be explained later in the chapter.