The binomial and Poisson distributions 559
Now try the following exercise
Exercise 213 Further problems on the
binomial 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, determine 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 (b) three
employees will be absent on a particular day.
[(a) 0.7514 (b) 0.0019]
4. A manufacturer estimates that 3% of his output
of a small item is defective. Find the probabilities 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]
5. Five coins are tossed simultaneously. Determine 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
⎤
⎥
⎥
⎥
⎦
6. If the probability of rain falling during a particular 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
⎤
⎥
⎥
⎥
⎥
⎥
⎦
7. An automatic machine produces, on average, 10% of its components outside of the
tolerance required. In a sample of 10 components from this machine, determine the probability of having three components outside of
the tolerance required by assuming a binomial
distribution.
[0.0574]
57.2 The Poisson distribution
When the number of trials, n, in a binomial distribution
becomes large (usually taken as larger than 10), the calculations associated with determining the values of the
terms becomes laborious. If n is large and p is small,
and the product np is less than 5, a very good approximation to a binomial distribution is given by the corresponding Poisson distribution, in which calculations
are usually simpler.
The Poisson approximation to a binomial distribution
may be defined as follows:
‘the probabilities that an event will happen 0, 1, 2,
3, … , n times in n trials are given by the successive
terms of the expression
e
−λ
1 + λ +
λ 2
2!
+
λ 3
3!
+ · · ·
taken from left to right’.
The symbol λ is the expectation of an event happening
and is equal to np.
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 product np is
80 × 0.03, i.e. 2.4, which is less than 5.
Hence a Poisson approximation to a binomial distribution 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 expression
e
−λ
1 + λ +
λ 2
2!
+
λ 3
3!
+ · · ·
Now try the following exercise
Exercise 213 Further problems on the
binomial 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, determine 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 (b) three
employees will be absent on a particular day.
[(a) 0.7514 (b) 0.0019]
4. A manufacturer estimates that 3% of his output
of a small item is defective. Find the probabilities 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]
5. Five coins are tossed simultaneously. Determine 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
⎤
⎥
⎥
⎥
⎦
6. If the probability of rain falling during a particular 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
⎤
⎥
⎥
⎥
⎥
⎥
⎦
7. An automatic machine produces, on average, 10% of its components outside of the
tolerance required. In a sample of 10 components from this machine, determine the probability of having three components outside of
the tolerance required by assuming a binomial
distribution.
[0.0574]
57.2 The Poisson distribution
When the number of trials, n, in a binomial distribution
becomes large (usually taken as larger than 10), the calculations associated with determining the values of the
terms becomes laborious. If n is large and p is small,
and the product np is less than 5, a very good approximation to a binomial distribution is given by the corresponding Poisson distribution, in which calculations
are usually simpler.
The Poisson approximation to a binomial distribution
may be defined as follows:
‘the probabilities that an event will happen 0, 1, 2,
3, … , n times in n trials are given by the successive
terms of the expression
e
−λ
1 + λ +
λ 2
2!
+
λ 3
3!
+ · · ·
taken from left to right’.
The symbol λ is the expectation of an event happening
and is equal to np.
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 product np is
80 × 0.03, i.e. 2.4, which is less than 5.
Hence a Poisson approximation to a binomial distribution 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 expression
e
−λ
1 + λ +
λ 2
2!
+
λ 3
3!
+ · · ·