251veeble
12/14/06 (Open in print layout)
The Great Veeblefetzer Problem
As everyone knows, a Veeblefetzer has two components, a Phillinx and a Flubberall. Depending
on the design of the Veeblefetzer, the Veeblefetzer will work a) as long as both components work or b) as
long as either component works.
Define the following events.
Period
Phillinx
Flubberall
Fails
Fails
1
A1
B1
2
A2
B2
3
A3
B3
And, for the moment, assume that the failure probabilities are as follows.
Period
1
2
3
Sum
Event
Phillinx
Fails
.5
.4
.1
1.0
B1
.30
Flubberall
Fails
.6
.2
.2
1.0
B2
Make some tables. The first is a joint
probability table. Assume that the events
involving the Phillinx are independent of
events involving the Flubberall.
B3
Sum
A1
A2
____
A3
Sum
1.00
a) Assume that the Veeblefetzer will work as long as both components work. Fill in the table with the
period in which the Veeblefetzer will fail.
Event
Sum
B1
B2
B3
Period 1
Period 1
A1
A2
A3
Sum
Use these two tables to figure out the probability that the Veeblefetzer will fail in each period.
Period
Component Joint Events
Probability
1
2
3
_______
Total
Can any of the results for a given period be expressed as a simple union or intersection of two events like
A1  B2 or A1  B2 ? If the second is true for any pair of events, show that the addition rule applies. If you
want to learn to use the equation writer in Word, click on A1  B2 and see what happens. If it is not
installed, it can usually be found on the Word disk.
b) Assume that the Veeblefetzer will work as long as one component works. Fill in the table with the period
in which the Veeblefetzer will fail.
Event
Sum
B1
B2
B3
Period 1
Period 2
A1
A2
A3
Sum
Use these two tables to figure out the probability that the Veeblefetzer will fail in each period.
Period
Component Joint Events
Probability
1
2
3
_______
Total
Can any of the results for a given period be expressed as a simple union or intersection of two events like
A1  B2 or A1  B2 ? If the second is true for any pair of events, show that the addition rule applies.
c) Assume that the Period 1 is time (year) 0 to 5, Period 2 is time 5 to 10 and period 3 is time 10 to 15.
Find the probability of a breakdown for each period if the life of the Flubberall has a continuous uniform
distribution between zero and 12 and the life of the Phillinx has a continuous uniform distribution with a
distribution between 2 and 15. Use these probabilities to rewrite the joint probability table and repeat b) and
c).
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