Probability and Statistics Problems

This page contains solutions to various probability and statistics problems including multiple choice exams, uniform distribution, and manufacturing process reliability.

Q U E S T I O N 1
1. Consider a simple experiment that is replicated a number of times. Consider four events
A, B, C, D. In this experiment events B and D are not independent and the only mutually
exclusive events are events A and B and events C and D . If the probability of event D
occurring is 0.42, the probability of event B occurring is 0.45 and the probability of event B
given D occurring together is 0.31, then what is the probability of event B AND D
occurring, P(B AND D)? Provide your answer to 3 decimal places.
8 points
Q U E S T I O N 2
1. A student is attempting a multiple choice exam consisting of 20 questions, with 1 mark
available for every correctly answered question and zero marks for an incorrect answer.
Each question has 5 possible answers to it, the student must select just 1 of these answers,
and for each question only 1 of the possible answers is correct. The student has not revised
for this exam and so decides to adopt the strategy of randomly guessing one of the 5
available options for each question.
Using this strategy and expressing probability as a number between 0 and 1, calculate to 3
decimal places (and never specify units or enter text of any kind):
The mark the student can expect to get is . (Hint: a student can expect to get the
average mark achieved by many students taking this exam).
The variance in the marks is .
The probability the student gets exactly 3 questions correct is .
The probability the student gets 10 or more questions correct is .
The number of ways the student can score 5 out of 20 for the exam is
For the student to expect to score zero marks by following his/her chosen strategy, a
negative mark per question of is needed.
20 points
Q U E S T I O N 3
1. A metal bolt has a diameter that has a uniform distribution. The mean diameter is 4.175 mm
with a variance of 0.001875.
Where a probability is asked for, do not express this as a percentage. Use fully un-rounded
values in all calculations, but provide answers to the following questions to 2 decimal
places and do not specify units or enter text of any kind:
The uniform parameter a is equal to mm.
The uniform parameter b is equal to mm.
The probability that the bolt will fit into a hole that has a diameter of 4.18 mm is .
0.560
4.000
3.200
0.205
0.003
0.175
-0.200
4.10
4.25
0.07

Suppose a bolt does fit into the hole with a diameter of 4.18 mm. The probability that the
difference between the diameter of the hole and the diameter of the bolt is less than or
equal to 0.005 mm equals .
The mean is and the variance is for the number of bolts in a box of 50
bolts with a diameter less than or equal to 4.19 mm.
20 points
Q U E S T I O N 4
1. Consider a manufacturing process composed of three stages - (R1, R2 and
R3). Stage one consists of 2 machines arranged in a doubly redundant sub
system. Stage two has 3 machines that work in series, meaning that this stage
operates successfully only if all the machines operate. Stage three is made up
of 4 machines and only operates successfully if at least 2 machines operate. The
three manufacturing stages are connected in series to produce the final product,
meaning that they must all work properly to produce a final product.
Between two scheduled maintenances, each machine may
fail independently of the others, with the following failure probabilities:
R1
0.18 0.22
R2
0.19 0.29 0.28
R3
0.35 0.24 0.19 0.38
Use this information to answer the following questions to 3 decimal
places (expressing probability as a number between 0 and 1 and do
not specify units or enter text of any kind ):
1. The reliability of sub system stage 1 is .
2. The reliability of sub system stage 2 is .
3. The reliability of sub system stage 3 is .
4. At stage 3 the probability of 2 machines breaking down between
scheduled maintenances is .
5. The probability of the manufacturing process as a whole failing
between scheduled maintenances is .
0.03
2.09 1.46
0.640
0.414
0.248
0.190
0.061