Tutorial 1(2) (Probability distributions)
An experiment in which satisfied the following characteristic is called a binomial experiment:
 The random experiment consists of n identical trials.
 Each trial can result in one of two outcomes, which we denote by success or failure.
 The trials are independent.
 p is the probability of success and the q  1  p is the probability of failure. p and q are constant
throughout the experiment.
Given that a random variable X is the number of successes, n is the number of independent trials, p is the
probability of success and q is the probability of failure. X is said to have binomial distribution,
X ~ Bn, p  with a probability function as follows:
n!
Note: n C x 
P( X  x)  nCx p x q n x x = 0, 1, 2, ......, n;
n  x ! x!
Mean,   E( X )  np ;
Variance,  2  V ( X )  np (1  p )  npq
1) Given that X ~ B5,0.4 . Find P X  3 , P X  3 , E X  and Var  X  .(0.2304, 0.3174, 2, 1.2)
2) Given that X ~ B10,0.2 . Find P X  0 , P X  1 , P X  2 , P X  2 , E X  and
Var  X  . (0.1074, 0.2684, 0.302, 0.6778, 2, 1.6)
3) Given that X ~ B25,0.15 . Find P X  5 , P X  7 , P5  X  8 and P2  X  10 .
(0.6821, 0.0255, 0.3099, 0.7442)
4) Suppose 20% of the marbles packed in a box are red in color. Suppose 4 marbles are
chosen at random. Find the probability that
a) Two are red (0.1536)
b) Three are red (0.0256)
c) None are red (0.4096)
5) Six fair coins are tossed. Let X denotes the number of “tails” occurring. Calculate mean
and variance of X. (3,1.5)
6) The probability that a certain machine will produce a defective item is 0.20. If a random
sample of 6 items is taken from the output of this machine, what is the probability that
there will be 5 or more defectives in the sample? (0.0016)
7) A statistics quiz consists of 10 multiple questions. There are four choices for each question.
One student is unprepared and decides to guess the answers to every question. Assuming
70% is a passing grade; find the probability that the student will pass the quiz. (0.0035)
8) 30% of pupils in a school travel to school by bus. From a sample of ten pupils chosen at
random, find the probability that
a) only three travel by bus (0.2668)
b) less than half travel by bus(0.8497)
9) Newsweek reported that 60% of young children have blood lead levels that could impair
their neurological development. Assuming that a class in a school is a random sample from
the population of all children at risk, the probability that at least 5 children out of 10 in a
sample taken from a school may have a blood level that may impair development is:(0 .84)
10) A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all
invoices, 10% receive the discount. In a company audit, 12 invoices are sampled at
random.
a) What is probability that fewer than 4 of 12 sampled invoices receive the
discount?(0.9774)
b) Then, what is probability that more than 1 of the 12 sampled invoices received a
discount. (0.341)
Given that a random variable X is the number of events occurring when the events occur in a continuum
of time or space. X is said to have poisson distribution, X ~ P0   with a probability function as
follows:
P( X  x) 
e   x
x!
x = 0, 1, 2, ......,
Mean,   E ( X )   ;
Variance,  2  V ( X )  
11) Let X ~ P0 4 . Find
a. P X  1 and P X  0 (0.0733, 0.0183)
b. P X  2 and P X  1 (0.0916, 0.9084)
c. E X  and Var X  (4, 4)
12) Let X ~ P0 12 . Find
a. P X  8 and P X  8 (0.1550, 0.0655)
b. P X  4 and P X  4 (0.9977, 0.9924)
c. P4  X  14 (0.7697)
13) If a keyboard operator averages 2 errors per page of newsprint, and if these errors follows
Poisson process, what is the probability that,
a) Exactly 4 erors will be found on a given page(0.0902)
b) At least 2 error will be found on a given page. (0.594)
14) On the average, 12 people per hour approach a decorating consultant with questions in a
fabric store. What is the probability that at least three people will approach the consultant
with questions during a 10-minute period? (0.3232)
15) The rate at which a particular defect occurs in lengths of plastic film being produced by a
stable manufacturing process is 4.2 defects per 75 meter length. A random sample of the
film is selected and it was found that the length of the film in the sample was 25 meters.
What is the probability that there will be at most 2 defects found in the sample?(0.8335)
16) Each 500-meter roll of sheet includes two flaws, on average. A flaw is a scratch which
would affect the use of that segment of sheet steel in the finished product. What is the
probability that a particular 100-meter segment will include no flaw? (0.6703)
17) The marketing manager of a company has noted that she usually receives 10 complaint
calls during a week (consisting of five working days), and that the calls occur at random.
Let us suppose that the number of calls during a week follows the Poisson distribution.
The probability that she gets five such calls in one day is:(0.0361)
18) Suppose that 0.03% of plastic containers manufactured by a certain process have small
holes that render them unfit for use. Let X represent the number of containers in a random
sample of 10000 have this defect. Find
a) P X  3 (0.2240)
b) P X  2 (0.4232)
c) P1  X  4 (0.5974)
d) E x  , Var x  and standard deviation x (3, 3, 1.73)
19) The number of messages received by a computer bulletin board is a Poisson random
variable with a mean rate of 8 messages per hour.
a) What is the probability that five messages are receive in a given hour? (0.0916)
b) What is the probability that 10 messages are received in 1.5 hours? (0.1048)
c) What is the probability that fewer than three messages are received in one-half hour?
(0.0005)
20) The number of points scored by Team A in a basketball match is Poisson distributed with
mean, 𝜇. If the probability that the team does not score any point in a match is 0.09,
calculate
a) 𝜇(2.4)
b) The probability that the team scores at least 16 points in 4 matches. (0.0362)
A random variable X having a normal distribution with mean 𝜇 and standard deviation, 𝜎can be written as
X ~ N , with its density function:

f ( x) 

1
2
e  x 
2
2
/ 2 2

A random variable Z having a standard normal distribution with mean 𝜇 = 0 and standard deviation, 𝜎 =
1 can be written as Z ~ N 0,1 with its density function:
 
f ( z) 
1 z
e
2
2
/2

Any value X from a normally distributed population
equivalent standard normal value by
Z
X 
X ~ N ,   can be transformed into the
2

21) Determine the probability or area for the portions of the normal distribution described.
a) P0  Z  0.91
b) P1.01  Z  0
c) PZ  0.93
d) P 1.2  Z  1.61 e) P1.1  Z  1.56 (0.3186,0.3438,0.8238,0.8312,0.0763)
22) Suppose X is a normal distribution, N 70,4 . Find
a) P67  X  75
b) P71  X  76
c) P63  X  68
d) P X  74 (0.927,0.3072,0.1585,0.0228)
23) Suppose the test scores of 600 students are normally distributed with a mean of 76 and
standard deviation of 8. The number of students scoring between 70 and 82 is:(328)
24) Marks on a Chemistry test follow a normal distribution with a mean of 65 and a standard
deviation of 12. Approximately what percentage of the students have scores below
50?(11%)
25) The distribution of weights in a large group is approximately normally distributed. The
mean is 80 kg and approximately 68% of the weights are between 70 and 90 kg. The
standard deviation of the distribution of weights is equal to:(10)
26) The heights of boys at a particular age follow a normal distribution with mean 150.3 cm
and variance 25 cm. Find the probability that a boy picked at random from this age group
has height
a.
less than 153 cm
b.
more than 158 cm
c.
between 150 cm and 158 cm
d.
more than 10 cm difference from the mean height
27) The number of shirts sold in a week by a shop is normally distributed with a mean of
2080 and a standard deviation of 50. Estimate
a.
the probability that in a given week fewer than 2000 shirts are sold
b.
the number of weeks in a year that between 2060 and 2130 shirts are sold
c.
the least number n of shirts such that the probability that more than n are sold
in a given week is less than 0.02.
28) The random variable X is normally distributed with a mean of 45. The probability that X is
greater than 51 is 0.288. Find the standard deviation of the distribution.
29) The random variable X is distributed N (μ, σ2) and it is known that P(X > 80) = 0.00113
and P(X < 30) = 0.0287. Find the value of μ and σ.
30) The average time it takes a group of adults to complete a certain achievement test is 46.2
minutes. The standard deviation is 8 minutes. Assume the variable is normally distributed.
a. Find the probability that a randomly selected adult will complete the test in less than
43 minutes.
b. Find the probability that, if 50 randomly selected adults take the test, the mean time it
takes the group to complete the test will be less than 43 minutes.
c. Does it seem reasonable that an adult would finish the test in less than 43 minutes?
Explain.
d. Does it seem reasonable that the mean of the 50 adults could be less than 43 minutes?
31) At a large publishing company, the mean age of proofreaders is 36.2 years, and the
standard deviation is 3.7 years. Assume the variable is normally distributed.
a. If a proofreader from the company is randomly selected, find the probability that his
or her age will be between 36 and 37.5 years.
b. If a random sample of 15 proofreaders is selected, find the probability that the mean
age of the proofreaders in the sample will be between 36 and 37.5 years.
32) 10% of the chocolates produced in a factory are mis-shapes. A random sample of 1000
chocolates is taken. Find the probability that
a.
less than 80 are mis-shapes
b.
between 90 and 115 (inclusive) are mis-shapes
c.
120 or more are mis-shapes
33) The number of calls per hour received by an office switchboard follows a Poisson
distribution with parameter 30. Using the normal approximation to the Poisson
distribution, find the probability that, in one hour
a.
there are more than 33 calls
b.
c.
there are between 25 and 28 calls (inclusive)
there are 34 calls
26. a. 0.7054 b. 0.0618
c. 0.4621
d. 0.0228
27. a. 0.0548 b. 26 c. 2183
28. 10.7
29. 49.2036, 10.1071
30. a. 0.3446 b. 0.0023
c. Yes d. Very unlikely
31. a. 0.1026 b. 0.4961
32. a. 0.0154 b. 0.7904
c. 0.02
33. a. 0.2614 b. 0.1865
c. 0.0559