Associated Topics || Dr. Math Home || Search Dr. Math

### Probability for a Given Distribution of Objects

```
Date: 7/24/96 at 16:4:8
From: Ariel Di Miro
Subject: Probability for a Given Distribution of Objects

1) A man has a box of light bulbs to sell. 80 percent of these bulbs
are class 'A', and 20% are class 'B'. The 'A' bulbs have a lifetime of
2400 hours, while 'B' bulbs last 2600 hours. The man takes a new bulb
from the box and uses it in his shop for 600 hours. Someone wants to
buy a bulb, but there are no more, so the man offers him the bulb he
has used in his shop. Calculate the probability that the bulb will
burn out in more than 2400 hours. (I think this problem needs more
data to be solved.)

2) The sizes of some screws are distributed "normally" (a Gaussian
distribution). They must be between: 10 +- 3 mm (9,7 and 10,3 mm). A
lot of them are tested with an instrument, and it's found that 5
percent of them are below the range, and 11 percent are above the
range (these are refused).

After the test, it is proved that the instrument made a systematic
error of +0.05 mm. What is the percentage of refused screws if the
test instrument was right?

3) A company makes wires. A failure in the machine made aleatory
mistakes in the wires, with an average frequency of 1 mistake in 25
meters. 30 wires with a length of 100 meters are selected. Which is
the probability that at least 3 wires have no more than 2 mistakes ?
(I think I have to use Poisson and Bernoulli.)

I would thank you very much if you help me to solve them.
```

```
Date: 7/26/96 at 8:50:44
From: Doctor Anthony
Subject: Re: Probability for a Given Distribution of Objects

(Question 1)
If the life-times have a mean of 2400 hours and 2600 hours, we still
need to know the standard deviations, if we are to calculate
probabilities of the bulb from the shop lasting a further 2400 hours.
Check the question again and see if the standard deviations are given.

(Question 2)
Because of the instrument error, the tail areas are 5 percent below
9.75 mm, and 11 percent above 10.35 mm. From this information we can
find the true mean and s.d. of the screws produced.  Let m = mean
and s = s.d. From tables a 5 percent tail below the mean corresponds
to z = -1.645   The tail area of 11 percent above the mean corresponds
to z = 1.2263

We have two equations using z = (x-m)/s for the two values of z and x
that we know.

-1.645 = (9.75 - m)/s

1.2263 = (10.35 - m)/s

-1.645s = 9.75 - m
1.2263s = 10.35 - m

Subtracting these equations we get  2.8713s = 0.6
s = 0.20896

Then from first equation  m = 9.75 + 1.645*0.20896
= 10.0937 mm

The percentage that are actually too small is therefore found from

z = (9.7 - 10.0937)/0.20896

= -1.8843 and from Normal tables area = .9702
and tail area = 1 - .9702 = 0.0298
= 2.98 percent

The percentage that are actually too large is found from

z = (10.3 - 10.0937)/0.20896

= 0.9873 and from Normal tables area = .9382
and tail area = 1 - .9382 = 0.0618
= 6.18 percent

Total percentage rejected = 2.98 + 6.18 = 9.16 percent

(Question 3)
We use the Poisson probability model to find probability of 0, 1, 2
etc. mistakes in 100 metres.  Because of the additive property of the
means of Poisson probabilities we can say that the mean number of
mistakes in 100 metres will be 4.

Poisson probabilities are P(0) = e^(-4)     = .018316
P(1) = (4/1)*P(0) = .073263
P(2) = (4/2)*P(1) = .146525
P(3) = (4/3)*P(2) = .195367
P(4) = (4/4)*P(3) = .195367  and so on

We need P(0) + P(1) + P(2) = .238104  and this is the probability that
one 100 metre wire has no more than 2 mistakes.

Now with 30 wires, at least 3, could mean 3, 4, 5,.... 30, so instead
we consider 1 - Prob(0 or 1 or 2) with no more than 2 mistakes.

For the binomial probabilities which we now consider

p = .238104   q = .761896  n = 30

P(0) = .761896^30 = .000286
P(1) = 30*.238104*.761896^29 = .002684
P(2) = 435*.238104^2*.761896^28 = .012165

P(0) + P(1) + P(2) = 0.015135  and subtracting this from 1 we get
0.984865

So the probability of at least 3 wires with no more than 2 mistakes is
0.984865

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Probability
College Statistics

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search