CHAP. 7] PROBABILITY 143
REPEATED TRIALS, BINOMIAL DISTRIBUTION
7.23. Suppose that, whenever horsesa,b,c,drace together, their respective probabilities of winning are 0.2,
0.5, 0.1, 0.2. That is,S={a, b, c, d}whereP(a)= 0 .2,P(b)= 0 .5,P(c)= 0 .1,P(d)= 0 .2. They race
three times.(a) Describe and find the number of elements in the product probability spaceS 3.
(b) Find the probability that the same horse wins all three races.
(c) Find the probability thata,b,ceach win one race.For notational convenience, we writexyzfor(x,y,z).(a) By definition,S 3 =S×S×S={xyz|x, y, z∈S}andP(xyz)=P(x)P(y)P(z).
Thus, in particular,S 3 contains 4^3 =64 elements.
(b) We seek the probability of the eventA={aaa,bbb, ccc, ddd}. By definition,P (aaa)=( 0. 2 )^3 = 0. 008 , P (ccc)=( 0. 1 )^3 = 0. 001
P (bbb)=( 0. 5 )^3 = 0. 125 , P(ddd)=( 0. 2 )^3 = 0. 008ThusP (A)= 0. 0008 + 0. 125 + 0. 001 + 0. 008 = 0 .142.
(c) We seek the probability of the eventB={abc, acb, bac, bca, cab, cba}. Every element inBhas the same
probability, the product( 0. 2 )( 0. 5 )( 0. 1 )= 0 .01. ThusP(B)= 6 ( 0. 01 )= 0 .06.7.24. The probability that John hits a target isp=^14. He firesn=6 times. Find the probability that he hits the
target: (a) exactly two times; (b) more than four times; (c) at least once.
This is a binomial experiment withn=6,p=^14 , andq= 1 −p=^34 ; that is,B( 6 ,^14 ). Accordingly, we use
Theorem 7.7.(a) P( 2 )=(
6
2)(
1
4) 2 ( 3
4) 4
= 15 ( 34 )/( 46 )=^12154096 ≈ 0 .297.(b) P( 5 )+P( 6 )=(
6
5)(
1
4) 5 ( 3
4) 1
+( 1
4) 6
=^1846 +^146 =^1946 = 409619 ≈ 0 .0046.(c) P( 0 )=( 3
4) 6
= 4096729 ,soP(X > 0 )= 1 − 4096729 =^33674096 ≈ 0 .82.7.25.A family has six children. Find the probabilitypthat there are: (a) three boys and three girls; (b) fewer
boys than girls. Assume that the probability of any particular child being a boy is^12.
Heren=6 andp=q=^12.(a) p=P(3 boys)=(
6
3)(
1
2) 3 (
1
2) 2
=20
64
=5
16
.(b) There are fewer boys than girls if there are zero, one, or two boys. Hence
p=P(0 boys)+P(1 boy)+P(2 boys)=(
1
2) 6
+(
6
1)(
1
2) 5
+(
6
2)(
1
2) 2 (
1
2) 4
=
11
32
= 0. 347.26.A man fires at a targetn=6 times and hits itk=2 times, (a) List the different ways that this can happen,
(b) How many ways are there?(a) List all sequences with twoS’s (successes) and fourF’s (failures):
SSFFFF, SFSFFF, SFFSFF, SFFFSF, SFFFFS, FSSFFF, FSFSFF, FSFFSF,
FSFFFS, FFSSFF, FFSFSF, FFSFFS, FFFSSF, FFFSFS, FFFFSS.(b) There are 15 different ways as indicated by the list. Observe that this is equal to(
6
2)
since we are distributing
k=2 lettersSamong then=6 positions in the sequence.