The binomial and Poisson distributions 559
Now try the following exercise
Exercise 213 Further problems on the
binomial distribution- Concrete blocks are tested and it is found
that, on average, 7% fail to meet the required
specification. For a batch of 9blocks, deter-
mine theprobabilitiesthat (a) three blocks and
(b) less than four blocks will fail to meet the
specification. [(a) 0.0186 (b) 0.9976] - 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 9blocks.
[(a) 0.2316 (b) 0.1408] - The average number of employees absent
from a firm each day is 4%. An office within
the firm has seven employees. Determine the
probabilitiesthat(a)noemployeeand(b)three
employees will be absent on a particular day.
[(a) 0.7514 (b) 0.0019] - Amanufacturerestimatesthat 3%ofhisoutput
of a small item is defective. Find the proba-
bilities that in a sample of 10 items (a) less
than two and (b) more than two items will be
defective. [(a) 0.9655 (b) 0.0028] - Five coins are tossed simultaneously. Deter-
mine the probabilities of having 0, 1, 2, 3, 4
and 5 heads upwards, and draw a histogram
depicting the results.
⎡
⎢
⎢
⎢
⎣
Vertical adjacent rectangles,
whose heights are proportional to
0.0313, 0.1563, 0.3125, 0.3125,
0.1563 and 0.0313⎤
⎥
⎥
⎥
⎦- If the probability of rain falling during a par-
ticular period is 2/5, find the probabilities of
having 0, 1, 2, 3, 4, 5, 6 and 7 wet days in a
week. Show these results on a histogram.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Vertical adjacent rectangles,
whose heights are proportional
to 0.0280, 0.1306, 0.2613,
0.2903, 0.1935, 0.0774,
0.0172 and 0.0016
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦- An automatic machine produces, on aver-
age, 10% of its components outside of the
tolerance required. In a sample of 10 compo-
nents from this machine, determine the prob-
ability of having three components outside of
the tolerance required by assuming a binomial
distribution. [0.0574]57.2 The Poissondistribution
When the number of trials,n, in a binomial distribution
becomes large (usually taken as larger than 10), the cal-
culations associated with determining the values of the
terms becomes laborious. Ifnis large andpis small,
and the productnpis less than 5, a very good approx-
imation to a binomial distribution is given by the cor-
responding Poisson distribution, in which calculations
are usually simpler.
The Poissonapproximationto a binomial distribution
maybedefinedasfollows:‘the probabilities that an event will happen 0, 1, 2,
3,...,ntimes inntrials are given by the successive
terms of the expressione−λ(
1 +λ+λ^2
2!+λ^3
3!+···)taken from left to right’.The symbolλis the expectation of an event happening
and is equal tonp.Problem 6. If 3% of the gearwheels produced
by a company are defective, determine the
probabilities that in a sample of 80 gearwheels
(a) two and (b) more than two will be defective.The sample number,n, is large, the probability of a
defective gearwheel,p, is small and the productnpis
80 × 0 .03, i.e. 2.4, which is less than 5.
Hence a Poisson approximation to a binomial dis-
tribution may be used. The expectation of a defective
gearwheel,λ=np= 2. 4
The probabilities of 0, 1 , 2 ,...defective gearwheels
are given by the successive terms of the expressione−λ(
1 +λ+λ^2
2!+λ^3
3!+···)