Probability & Statistics: Solved Assignment Problems

This assignment consists of several probability and statistics problems. The first question involves calculating the probability of the intersection of two non-independent events. The second question deals with a multiple-choice exam scenario, focusing on expected marks, variance, probability of specific scores, and the impact of negative marking. The third question explores a uniform distribution related to metal bolt diameters, calculating probabilities and parameters. The final question involves a manufacturing process with multiple stages and machines, assessing system reliability and failure probabilities. Each problem requires precise calculations and understanding of statistical concepts, with final answers rounded to specified decimal places. Desklib provides solved assignments and past papers for students.

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