ES 303 STATISTICAL METHODS FOR ENGINEERS
Assignment 1 (Section: 3 and 4)
Due date: December 1, 2024, till 10.00 (a.m.) (You should take photo of your solutions via phone or
scanner and then upload to assignment link at METU CLASS as a single file). File name should be
appropriate (Student Name-Assignment1). Make sure that uploaded pages are readable and the file is in
pdf or jpeg format. Solutions for the questions will be announced at 10.10, Therefore no late submission
will be possible and acceptable.
Questions
1) Consider a bicyclist who leaves at point Q (Figure 1-given below), choosing one of the roads randomly. The
probabilities for choosing the roads are given on each line (for the paths between letters).
A) What is the probability that he will not choose road QR1?
B) What is the probability for choosing road from R3 to A?
C) What is the probability that he will arrive at point A.
D) If he arrived at point A what is the probability that this was via road QR3? (Conditional Probability)
Figure 1.
2) Given that f(x)= k (7-x) for x= 1, 2, 3, 4, 5 is a probability function of a discrete random variable
Find a) k b) probability that X is at most 3, c) E[X], d) E[X2] e) E[3X-5], f) Var(X), g) x, h) Var(2X+6)
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3) A firm is placing 3 different orders for supplies to five different distributors. Each order is randomly
assigned to one of the distributors, and a distributor can recieve multiple orders.
Find the probability that a) all orders go to different distributors, b) all orders go to the same distributor, c)
exactly 2 of the 3 orders go to one particular distributor.
4) Four boxes, B1, B2, B3, B4 contain fuses. The boxes have 400, 300, 200 and 100 fuses respectively. The
percentages of defective fuses in the boxes are 3%, 2%, 1% and 1 %, respectively. All fuses were mixed in a
lot and one fuse is selected at random,
a) What is the probability that the selected fuse is defective?
b) What is the probability that the selected fuse is nondefective?
c) If the selected fuse is defective what is the probability that it was initially from Box B2?
d) If the boxes were not mixed initially and one fuse was selected at random, arbitrarily, from one of
the boxes what would be the probability that selected fuse is defective?
5) An experiment has only 4 possible mutually exclusive outcomes; A, B, C, D. Check whether the following
assignments of probabilities are permissible.
a) P[A]=0.32, P[B]=0.12, P[C]=0.28, P[D]=0.30
b) P[A]=1/3, P[B]=1/4, P[C]=1/6, P[D]=1/4
6) The following frequency table shows the classification of landfills in a region according to their concentration
of the hazardous chemicals; arsenic, barium and mercury. If a landfill is selected at random,
a) find the probability that it has a low concentration of mercury,
b) what is the probability that it has a high concentration of mercury and low concentration of arsenic and low
concentration of barium?
c) find the probability that it has a high concentrations of any two of the chemicals and low concentration of
the third one
d) are the events “high concentration of barium” and “high concentration of arsenic” dependent or independent?
e) Given that a landfill (selected at random) is found to have a high concentration of arsenic, what is the
probability that its concentration is high in mercury?
Barium
High
Arsenic Low
High
Low
Mercury
Mercury
High
2
Low
4
High
6
Low
11
1
7
9
14
2
7) A geological study indicates that an exploratory oil well drilled in a certain region should strike oil with
probability 0.35. (Assume independent and identical trials for this experiment). Find the probability that
a) The first strike of oil comes on the 5th well drilled
b) Second strike of oil comes on the 6th well drilled.
c) Find the expected value and variance of the number of wells that must be drilled to find 3rd
successful well (with strike of oil).
8) The probability that a certain wide-flange column will fail under a given axial load is 0.25. Assume Bernoulli
conditions are valid for this case. Use Binomial Distribution table below for answers. What are the probabilities
that among 13 such columns
a) At least 5 fails
b) Exactly 8 fails
c) Between 2 and 5 (inclusively) fail
d) What is the expected number of column that do not fail?
9) The number of flaws in a fiber optic cable follows a Poisson distribution with an average of 0.2 per 100 feet.
a) Find the probability of exactly 4 flaws in a 500 feet cable. b) Find the probability of exactly one flaw in the
first 300 feet and exactly one flaw in the next 400 feet.
10) Among the 20 items produced by a plant, 16 are flat-plate collectors and the others are concentrating
collectors. If a person visiting the show randomly selects 5 of the solar collectors to check out, what is the
probability that A) 4 of them will be flat plate collectors? B) all of them are concentrating collectors (explain
you answer)?
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