Higher Engineering Mathematics

558 STATISTICS AND PROBABILITY

0123456

Number of people

0.04

0.08

0.12

0.16

0.20

0.24

0.28

Probability of having an accident

0

Figure 57.2

Now try the following exercise.

Exercise 215 Further problems on the

Poisson distribution

1. In problem 7 of Exercise 214, page 556,
   determine the probability of having three
   components outside of the required tolerance
   using the Poisson distribution. [0.0613]


2. The probability that an employee will go to
   hospital in a certain period of time is 0.0015.
   Use a Poisson distribution to determine the
   probability of more than two employees
   going to hospital during this period of time
   if there are 2000 employees on the payroll.
   [0.5768]


3. When packaging a product, a manufacturer
   finds that one packet in twenty is under-
   weight. Determine the probabilities that in a

box of 72 packets (a) two and (b) less than

four will be underweight.

[(a) 0.1771 (b) 0.5153]

4. A manufacturer estimates that 0.25% of his
   output of a component are defective. The
   components are marketed in packets of 200.
   Determine the probability of a packet con-
   taining less than three defective components.
   [0.9856]


5. The demand for a particular tool from a
   store is, on average, five times a day and the
   demand follows a Poisson distribution. How
   many of these tools should be kept in the
   stores so that the probability of there being
   one available when required is greater than
   10%?⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

The probabilities of the demand

for 0, 1, 2,...tools are

0.0067, 0.0337, 0.0842, 0.1404,

0.1755, 0.1755, 0.1462, 0.1044,

0.0653,...This shows that the

probability of wanting a tool

8 times a day is 0.0653, i.e.

less than 10%. Hence 7 should

be kept in the store

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

6. Failure of a group of particular machine
   tools follows a Poisson distribution with a
   mean value of 0.7. Determine the probabil-
   ities of 0, 1, 2, 3, 4 and 5 failures in a week
   and present these results on a histogram.⎡

⎢

⎢

⎣

Vertical adjacent rectangles

having heights proportional

to 0.4966, 0.3476, 0.1217,

0.0284, 0.0050 and 0.0007

⎤

⎥

⎥

⎦