Exercise #4:
Part A
A company manufactures motor starters, of which 6% are defective. Each
starter is subjected to a control unit whose reliability is not perfect. This unit of
control rejects 5% of starters working properly. In addition, 0.12% of starters
are accepted for inspection and are defective.
It is noted that:
●
D the event: "the starter is defective">;
R the event: «the control unit rejects the starter».
1) (a) Translate the data from the statement using probability ratings.
b) Demonstrate that PD(R bar) = 0.02.
c) Construct the weighted tree corresponding to this situation.
2) a) Demonstrate that P(R) = 0.1058.
b) Is it safe to say that there is at least a 43% chance that a starter rejected at
control works properly? The answer will be justified.
Part B
A salesman of this company is in charge of selling lots of starters every day
working week. His boss has set him a goal that he will try to meet. Each
evening the assessment is made to know if the work was done properly, with possibly
an advice session.
We note the event "The objective was reached on the second day" and Pn = P(0n)
The first day, being a beginner, the salesman has only 1 chance in 5 to reach his goal.
Thus p = 0.2
We find that when one day the goal is reached, the probability for it to be
next day is 0.9.
On the other hand if one day the goal is not reached, the probability that it is not the next day
is 0.6.
Calculate P(0 ) and check that P(O ) =P(0 bar).
2) Show that Pn+1 = 0.5pm + 0.4
3) Pose vn = Pn - 0.8
a) Show that the sequence (vn)n21 is a geometric sequence which will be determined
first term and reason.
b) Determine the general term for the suite (vn) and then for the suite (Pn).
c) Determine the limit pn when n tends to + infinity. Interpret this result.