This document contains information about several probability problems: 1) A study analyzed scratch and shock resistance of discs. It provides the probabilities of discs having high/low scratch and shock resistance. It asks to calculate several probabilities based on this data. 2) A box contains 7 balls, 3 are black and 4 are white. It asks for the probability the second ball drawn is black, and the conditional probability the first is black given the second is black. 3) It provides information about the health of 22 players on a team and asks for the probability a randomly selected player has various conditions. 4) It provides information about vegetables being graded on various qualities and asks for probabilities related to these grades.

1. Tutorial 1
1. The compact discs from a certain supplier are analyzed for scratch and shock resistance.
The results from 100 discs tested are summarized as follows:
Scratch Resistance
High Low
Shock
Resistance
High 30 10
Medium 22 8
Low 25 5
Let A denote the event that a disc has high shock resistance, and B denote the event
that a disc has high scratch resistance. If sample is selected at random, determine the
following probabilities:
(a) P(A) (b) P(B) (c) P(B’) (d) P(AUB) (e) P(A∩ B)
(f) P(AUB’) (g) )( BAP (h) )( ABP
2. A box contains 3 black and 4 while balls. Two balls are drawn at random one at a time
without replacement.
(i) What is the probability that a second ball drawn is black?
(ii) What is the conditional probability that first ball drawn is black if the second ball
is known to be black?
3. In a group of 22 players of Harimau Malaya team, only 7 of them are totally fit for the
Asian game tournament and every person of the remaining 15 players has either knee
injury or flue fever or both. However 8 of the players have knee injuries and 10 have flu
fever. Let A be the event of knee injuries and B be the event of flu fever. If one of them
selected randomly from this group, what is the probability that he has
i. both, knee injury and high flu fever?]
ii. knee injury only?
iii. flu fever only?
iv. either knee injury or flu fever?
4. An agricultural research establishment grows vegetables and grades each one as either
good or bad for taste, good or bad for its size, and good or bad for its appearance.
Overall, 78% of the vegetables have a good taste. However, only 69% of the vegetables
have both a good taste and a good size. Also, 5% of the vegetables have a good taste
and a good appearance, but a bad size. Finally, 84% of the vegetables have either a good
size or a good appearance.

2. i. if a vegetable has a good taste, what is the probability that it also has a good
size?
ii. if a vegetable has a bad size and a bad appearance, what is the probability that it
has a good taste?
5. Three machines A, B and C produce identical items of their respective output 5%, 4%
and 3% of the items are faulty. On a certain day A has produced 25%, B has produced
30% and C has produced 45% of the total output. An item selected at random is found
to be faulty. What are the chances that it was produced by C?
6. Suppose that a test for Influenza A, H1N1 disease has a very high success rate: if a tested
patient has the disease, the test accurately reports this, a ’positive’, 99% of the time,
and if a tested patient does not have the disease, the test accurately reports that, a
’negative’, 95% of the time. Suppose also, however, that only 0.1% of the population
have that disease.
(i) What is the probability that the test returns a positive result?
(ii) If the patient has a positive, what is the probability that he has the disease?
(iii) What is the probability of a false positive?
oooOOOooo
RR/Tutorial Part 1/May 2015.

3. Tutorial 2
1. Find the mean, median and mode for the following observations:
2.3 3.6 2.6 2.8 3.2 3.6 4.3 5.2 6.9 2.8 3.6
2. The following data are direct solar intensity measurements (watts/m2
) on different days
at a location in southern Spain: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878,
909, 918, 558, 768, 870, 918, 940, 946, 661, 820, 898, 935, 952, 957, 693, 835, 905, 939,
955, 960, 498, 653, 730, 753. Calculate the sample mean, sample variance and sample
standard deviation.
3. The cold start ignition time of an automobile engine is being investigated by a
gasoline manufacturer. The following times (in seconds) were obtained: 1.99,
3.13, 3.26, 2.65, 0.85, 2.87, 5.96, 2.46, 1.89 and 3.35.
i. Calculate the sample mean, sample variance and sample standard
deviation.
ii. Construct a box-plot of these data.
4. The average age of the football players on each team of the premier league are as
follows.
29.4 29.8 29.4 31.8 32.7 34.0
28.5 27.9 30.9 29.3 28.8 28.6
29.1 31.0 30.7 30.3 29.7 31.0
28.4 28.9 27.7 28.7 30.5 29.8
26.6 27.9 27.9 29.9 29.3 28.1
i. Construct a stem-and-leaf display for these data;
ii. Find the mode and the mean of the data;
iii. Construct a box-plot for these data.
5. The following data are the joint temperatures of the O-rings (°F) for each test firing or
actual launch of the space shuttle rocket motor (from Presidential Commission on the
Space Shuttle Challenger Accident, Vol. 1, pp. 129–131): 84, 49, 61, 40, 83, 67, 45, 66,
70, 69, 80, 58, 68, 60, 67, 72, 73, 70, 57, 63, 70, 78, 52, 67, 53, 67, 75, 61, 70, 81, 76, 79,
75, 76, 58, 31.
i. Calculate the median, the quartiles and the IQR;
ii. Construct a box plot of the data and comment on the possible presence of
outliers.

4. Tutorial 3
1. At UTP, the business students run an investment club. Each semester they create investment
portfolios in multiples of RM1,000 each. Records from the past several years show the
following probabilities of profits (rounded to the nearest RM50). In the table below, x =
profit per RM1, 000 and P(x) is the probability of earning that profit.
x 0 50 10
0
15
0
200
0.1
5
0.3
5
k 0.2 0.0
5
i. Determine the value of k that results in a valid probability distribution.
ii. The profit per RM1, 000 is a random variable. Is it discrete or continuous?
Explain.
iii. Find the expected value of the profit in a $1,000 portfolio.
iv. Find the standard deviation of the profit.
v. What is the probability of a profit of $150 or more in a RM1, 000 portfolio?
2. The output of a chemical process is continually monitored to ensure that the
concentration remains within acceptable limits. Whenever the concentration drifts
outside the limits, the process is shut down and recalibrated. Let x be the number of
times in a given week that the process is calibrated. The probability function f (x) is given
below.
x 0 1 2 3 4
f (x) 0.17 0.36 k 0.13 0.03
i. Is this a continuous or discrete distribution?
ii.Determine the value of k that results in a valid probability distribution.
iii. Find the mean of x.
iv. Find the variance of x .
v.What is the probability that the process is recalibrated fewer than two
times during a week?

5. 3. Let X be a continuous random variable with pdf given by


 <<−−
=
elsewhere,0
11,)1(
)(
xxk
xf
Find
i. the value of constant k;
ii. P(X < 0.5);
iii. the mean of X;
iv. the standard deviation of X;
v. cumulative distribution function of X.
4. Let X be continuous random variable with the following distribution
+∞<<∞−




 −
−= x
x
xf ,
8
)2(
exp
8
1
)(
2
π
i. What are the mean and the variance of X without calculation?
ii. What is the transformation needed to change the mean of X to 0 and the variance
of X to 1?
iii. Without calculation, find the value of .
2
exp
2
∫
+∞
∞−






− dx
x
5. Let X be a continuous random variable with pdf given by


 ≤≤
=
elsewhere,0
30,
)(
2
xkx
xf
Find
i. the value of constant k;
ii. the cdf, F(x);
iii. P(X >1)
iv. the mean of X; and
v. the variance of X.
oooOOOooo

6. RR/Tutorial Part 1/May 2015.
Tutorial 4
1. The probability of successfully landing a plane using a flight simulator is given as 0.70.
Nine randomly and independently chosen student pilots are asked to try to fly the plane
using the simulator.
i. What is the probability that all the student pilots successfully land the plane using
the simulator?
ii. What is the probability that none of the student pilots successfully lands the plane
using the simulator?
iii. What is the probability that exactly eight of the student pilots successfully land the
plane using the simulator?
2. At the Mc Donald drive-thru window of food establishment, it was found that during
slower periods of the day, vehicles visited at the rate of 15 per hour. Determine the
probability that
i. no vehicles visiting the drive-thru within a ten-minute interval during one of these
slow periods;
ii. only 3 vehicles visiting the drive-thru within a ten-minute interval during one of
these slow periods; and
iii. at least three vehicles visiting the drive-thru within a ten-minute interval during one
of these slow periods.
3. The number of cracks in a section of PLUS highway that are significant enough to require
repair is assumed to follow a Poisson distribution with a mean of two cracks per
kilometer. Determine the probability that
i. there are no cracks at all in 2km of highway;
ii. at least one crack in 500meter of highway; and
iii. there are exactly 3 cracks in 0.5km of highway.
4. Suppose a random variable, X has a uniform distribution with a = 5 and b = 9. Determine,
i. the pdf of X;
ii. the cdf of X;
iii. P(6 < X < 8).
iv. P(X < 7).
v. the mean of X; and
vi. the standard deviation of X.

7. 5. The time between telephone calls to ASTRO, a cable television payment processing
center follows an exponential distribution with a mean of 1.5 minutes. What is the
probability that the time between the next two calls
i. at least 45 seconds?
ii. will be between 50 to 100 seconds?; and
iii. at most 150 seconds?
6. An average LCD Projector bulb manufactured by the ABC Corporation lasts 300 days
with variance of 2500days. By assuming that the bulb life is normally distributed, what is
the probability that the bulb will last
i. at most 365 days?
ii. between 250days and 350days?
iii. at least 400days?
7. Given that X is normally distributed with mean 20 and standard deviation 2, compute
the following for n=40.
i. Mean and variance of X
ii. )19( ≤XP
iii. )22( >XP
iv. )5.2119( ≤≤XP
oooOOOooo
RR/Tutorial Part 1/May 2015.

8. Tutorial 5
1. Let X denote the number of flaws in a 1 in length of copper wire. The pmf of X is given in
the following table
X=x 0 1 2 3
P(X=x) 0.48 0.39 0.12 0.01
100 wires are sampled from this population. What is the probability that the average
number of flaws per wire in this sample is less than 0.5?
2. At a large university, the mean age of the students is 22.3 years, and the standard
deviation is 4 years. A random sample of 64 students is drawn. What is the probability
that the average age of these students is greater than 23 years?
3. Suppose that a sample of n = 1,600 tires of the same type are obtained at random from
an ongoing production process in which 8% of all such tires produced are defective.
What is the probability that in such a sample 150 or fewer tires will be defective?
4. Suppose that at a certain automobile plant the average number of work stoppages per
day due to equipment problems during the production process is 12.0.What is the
approximate probability of having 15 or fewer work stoppages due to equipment
problems on any given day?
5. The number of cars arriving per minute at a toll booth on a particular bridge is Poisson
distributed with a mean of 2.5.What is the probability that in any given minute
i. no cars arrive?
ii. not more than two cars arrive?
If the expected number of cars arriving at the toll booth per ten-minute interval is 25.0,
what is the approximate probability that in any given ten-minute period
iii. not more than 20 cars arrive?
iv. between 20 and 30 cars arrive?

9. 6. The overall length of a skew used in a knee replacement device is monitored using
and R charts. The following table gives the length for 20 samples of size 4.
(Measurements are coded from 2.00 mm; that is, 15 is 2.15 mm.
Observation Observation
Sample 1 2 3 4 Sample 1 2 3 4
1 16 18 15 13 11 14 14 15 13
2 16 15 17 16 12 15 13 15 16
3 15 16 20 16 13 13 17 16 15
4 14 16 14 12 14 11 14 14 21
5 14 15 13 16 15 14 15 14 13
6 16 14 16 15 16 18 15 16 14
7 16 16 14 15 17 14 16 19 16
8 17 13 17 16 18 16 14 13 19
9 15 11 13 16 19 17 19 17 13
10 15 18 14 13 20 12 15 12 17
i. Using all the data, find trial control limits for and R charts, construct the chart,
ii. and plot the data.
iii. Use the trial control limits from part (a) to identify out-of-control points. If
necessary, revise your control limits, assuming that any samples that plot outside
the control limits can be eliminated.
iv. Assuming that the process is in control, estimate the process mean and process
standard deviation.

10. 7. Astro Electronics division uses statistical quality control tools to ensure proper
weld strength in several of their welding procedures. Weld strength is measured
by a pull test to destruction. A sample of five items is taken periodically
throughout the production process. The average and the range of 22 samples are
listed and given in table below:
Sample
number
Sample
mean
Sample
range
Sample
number
Sample
mean
Sample
range
1
2
3
4
5
6
7
8
9
10
11
5.48
5.42
5.42
5.40
5.52
5.32
5.34
5.58
4.54
5.42
5.58
1.4
1.6
1.4
0.5
1.7
0.7
1.6
1.2
0.6
1.6
0.5
12
13
14
15
16
17
18
19
20
21
22
5.06
4.82
4.86
4.68
5.28
4.68
4.94
4.90
4.96
5.06
5.22
1.4
1.9
1.3
0.9
1.6
1.1
0.6
1.0
0.7
1.8
0.8
i. Use the given data to construct the X - chart and the R - chart.
ii. Interpret your results and comment on the stability of the process.
iii. Assuming that the process is in control, estimate the process mean and
process standard deviation.
oooOOOooo
RR/Tutorial Part 1/May 2015.