Higher Engineering Mathematics

THE BINOMIAL AND POISSON DISTRIBUTIONS 555

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The probability of there being 0, 1, 2,...,6
damaged components is given by the successive

terms of (q+p)^6 , taken from left to right.

(q+p)^6 =q^6 + 6 q^5 p+ 15 q^4 p^2 + 20 q^3 p^3 +···

(a) The probability of one damaged component is

6 q^5 p= 6 × 0. 925 × 0. 08 =0.3164

(b) The probability of less than three damaged com-
ponents is given by the sum of the probabilities
of 0, 1 and 2 damaged components.

q^6 + 6 q^5 p+ 15 q^4 p^2

= 0. 926 + 6 × 0. 925 × 0. 08

+ 15 × 0. 924 × 0. 082

= 0. 6064 + 0. 3164 + 0. 0688 =0.9916

Histogram of probabilities

The terms of a binomial distribution may be repre-
sented pictorially by drawing a histogram, as shown
in Problem 5.

Problem 5. The probability of a student suc-

cessfully completing a course of study in three

years is 0.45. Draw a histogram showing the

probabilities of 0, 1, 2,..., 10 students success-

fully completing the course in three years.

Letpbe the probability of a student successfully
completing a course of study in three years andqbe
the probability of not doing so. Thenp= 0 .45 and
q= 0 .55. The number of students,n,is10.
The probabilities of 0, 1, 2,..., 10 students suc-
cessfully completing the course are given by the
successive terms of the expansion of (q+p)^10 , taken
from left to right.

(q+p)^10 =q^10 + 10 q^9 p+ 45 q^8 p^2 + 120 q^7 p^3

+ 210 q^6 p^4 + 252 q^5 p^5 + 210 q^4 p^6

+ 120 q^3 p^7 + 45 q^2 p^8 + 10 qp^9 +p^10

Substitutingq= 0 .55 andp= 0 .45 in this expan-
sion gives the values of the successive terms as:
0.0025, 0.0207, 0.0763, 0.1665, 0.2384, 0.2340,
0.1596, 0.0746, 0.0229, 0.0042 and 0.0003. The
histogram depicting these probabilities is shown in
Fig. 57.1.

012345678910

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

Probability of successfully completing course

Number of students

0

Figure 57.1

Now try the following exercise.

Exercise 214 Further problems on the bino-

mial distribution

1. Concrete blocks are tested and it is found
   that, on average, 7% fail to meet the required
   specification. For a batch of 9 blocks, deter-
   mine the probabilities that (a) three blocks
   and (b) less than four blocks will fail to meet
   the specification. [(a) 0.0186 (b) 0.9976]


2. If the failure rate of the blocks in Problem 1
   rises to 15%, find the probabilities that (a) no
   blocks and (b) more than two blocks will
   fail to meet the specification in a batch of
   9 blocks. [(a) 0.2316 (b) 0.1408]


3. The average number of employees absent
   from a firm each day is 4%. An office within
   the firm has seven employees. Determine
   the probabilities that (a) no employee and