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ENGINEERING DATA ANALYSIS (MATH 4)
FOR BACHELOR OF SCIENCE IN CIVIL ENGINEERING (BSCE)
Probability
Probability is a field of mathematics that deals with chance. An experiment is an activity in which the
results cannot be predicted with certainty. Each repetition of an experiment is called a trial. An outcome is a
result of an experiment. An event is any collection of outcomes, and a simple event is an event with only one
possible outcome. An event can be one outcome or more than one outcome. For example, if a die is rolled
and a 6 shows, this result is called an outcome, since it is a result of a single trial. An event with one
outcome is called a simple event. The event of getting an odd number when a die is rolled is called a
compound event, since it consists of three outcomes or three simple events. In general, a compound
event consists of two or more outcomes or simple events. The sample space for a given experiment is a set
S that contains all possible outcomes of the experiment.
The sample space for the experiment of throwing a die and observing the number of dots on the top
face is the set { 1, 2, 3, 4,5, 6}.
In any experiment for which the sample space is S, the probability of an event occurring is given by the formula
𝑃 𝐸𝑣𝑒𝑛𝑡 = 𝑛
𝑛 𝐸𝑣𝑒𝑛𝑡
Sample Space
𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑝𝑎𝑐𝑒
Event
Tools in obtaining the total possible outcomes
1.
Tree Diagram
Example 1. Use a tree diagram to show all the possible outcomes when two unbiased coins are tossed.
Example 2.In a drawer there are some white socks and some black socks. Tim takes out one sock and then a
second. Draw a tree diagram to show the possible outcomes.
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2. Systematic Listing
Caitlin and Dave each buy a chocolate bar from a vending machine that sells Aero, Bounty, Crunchie and Dime
bars. List the possible pairs of bars which Caitlin and Dave can choose.
3. Tables
Example 1.A fair dice is rolled and an unbiased is tossed. Draw a table to show the possible outcomes.
Example 2. Draw a table to show all the possible total scores when two fair dice are thrown at the same time.
Example 3. Find the sample space for rolling two dice.
There are three basic interpretations of probability:
1. Classical probability
2. Empirical or relative frequency probability
3. Subjective probability
Classical Probability
Classical probability uses sample spaces to determine the numerical probability that an event
will happen. You do not actually have to perform the experiment to determine that probability. Classical
probability is so named because it was the first type of probability studied formally by mathematicians
in the 17th and 18th centuries.
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Classical probability assumes that all outcomes in the sample space are equally likely to occur.
For example, when a single die is rolled, each outcome has the same probability of occurring. Since there
are six outcomes, each outcome has a probability of .
Rounding Rule for Probabilities. Probabilities should be expressed as reduced fractions or rounded to
two or three decimal places. When the probability of an event is an extremely small decimal, it is
permissible to round the decimal to the first nonzero digit after the point. For example, 0.0000587
would be 0.00006.
Examples:
1. Find the probability of getting a black 10 when drawing a card from a deck.
2. If a family has three children, find the probability that two of the three children are girls.
3. A card is drawn from an ordinary deck. Find these probabilities.
a. Of getting a jack.
b. Of getting the 6 of clubs (i.e., a 6 and a club).
c. Of getting a 3 or a diamond.
d. Of getting a 3 or a 6.
FOUR BASIC PROBABILITY RULES
Examples:
1. When a single die is rolled, find the probability of getting a 9. ( Ans. Zero or Impossible)
2. When a single die is rolled, what is the probability of getting a number less than 7? (Ans. One or
Certain)
Empirical Probability
The difference between classical and empirical probability is that classical probability assumes
that certain outcomes are equally likely (such as the outcomes when a die is rolled), while empirical
probability relies on actual experience to determine the likelihood of outcomes. In empirical probability,
one might actually roll a given die 6000 times, observe the various frequencies, and use these
frequencies to determine the probability of an outcome.
Suppose, for example, that a researcher for the American Automobile Association
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(AAA) asked 50 people who plan to travel over the Thanksgiving holiday how they will get to their
destination. The results can be categorized in a frequency distribution as shown.
Example: In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2
had type AB blood. Set up a frequency distribution and find the following probabilities.
a. A person has type O blood.
b. A person has type A or type B blood.
c. A person has neither type A nor type O blood.
d. A person does not have type AB blood.
Example 2: Hospital records indicated that knee replacement patients stayed in the hospital for the
number of days shown in the distribution.
Find these probabilities.
a. A patient stayed exactly 5 days.
b. A patient stayed less than 6 days.
c. A patient stayed at most 4 days.
d. A patient stayed at least 5 days.
Subjective Probability
The third type of probability is called subjective probability. Subjective probability uses a
probability value based on an educated guess or estimate, employing opinions and inexact information.
In subjective probability, a person or group makes an educated guess at the chance that an event
will occur. This guess is based on the person’s experience and evaluation of a solution. For example, a
sportswriter may say that there is a 70% probability that the
Pirates will win the pennant next year. A physician might say that, on the basis of her diagnosis,
there is a 30% chance the patient will need an operation. A seismologist might say there is an 80%
probability that an earthquake will occur in a certain area. These are only a few examples of how
subjective probability is used in everyday life.
All three types of probability (classical, empirical, and subjective) are used to solve a variety of
problems in business, engineering, and other fields.
LET’S DO AN EXERCISE
1. A coin is tossed. Find:
a) the sample space.
b) the probability of getting a head.
2. Find the probability of getting at least two 4’s in a roll of three dice.
3. A number of five different digits is built at random from the digits (1, 2, 3, 4, 5, 6, 8). Find the probability
that the number will have even digits at each end.
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Addition Rules for Probability
Addition Rule 1. Probability of Mutually Exclusive Events- two or more events are mutually exclusive if the
occurrence of one of them excludes the probability of the others to happen in the same trial. It refers to as the
sum of their separate probabilities.
P (A U B) = P (A) + P (B)
Addition Rule 2. Not Mutually Exclusive- events occur together.
P( A or B)= P(A)+ P(B)-P(A and B)
Examples:
1. Determine which events are mutually exclusive and which are not, when a single die is rolled.
a. Getting an odd number and getting an even number
b. Getting a 3 and getting an odd number
c. Getting an odd number and getting a number less than 4
d. Getting a number greater than 4 and getting a number less than 4
2. Determine which events are mutually exclusive and which are not when a single card is drawn from a
deck.
a. Getting a 7 and getting a jack
b. Getting a club and getting a king
c. Getting a face card and getting an ace
d. Getting a face card and getting a spade
3.
A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a person selects a
doughnut at random, find the probability that it is either a glazed doughnut or a chocolate doughnut.
4. At a political rally, there are 20 Republicans, 13 Democrats, and 6 Independents. If a person is selected at
random, find the probability that he or she is either a Democrat or an Independent.
5. A single card is drawn at random from an ordinary deck of cards. Find the probability that it is either an
ace or a black card.
6. On New Year’s Eve, the probability of a person driving while intoxicated is 0.32, the probability of a person
having a driving accident is 0.09, and the probability of a person having a driving accident while
intoxicated is 0.06. What is the probability of a person driving while intoxicated or having a driving
accident?
The Multiplication Rules and Conditional Probability
Multiplication Rules can be used to find the probability of two or more events that occur in sequence.
Multiplication Rule 1. Two events A and B are independent events if the fact that A occurs does not affect the
probability of B occurring.
P A and B = P A •P B
Multiplication Rule 2. When the outcome or occurrence of the first event affects the outcome or occurrence of
the second event in such a way that the probability is changed, the events are said to be dependent events.
P A and B = P A •P B∣A .
Examples:
1. A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the
die.
2. A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of
getting a queen and then an ace.
3. Example: Three cards are drawn from an ordinary deck and not replaced. Find the probability of
these events.
i) Getting 3 jacks.
ii) Getting an ace, a king, and a queen in order.
iii) Getting a club, spade, and a heart in order.
iv) Getting 3 clubs.
Conditional Probability
The probability that the second event B occurs given that the first event A has occurred can be found by
dividing the probability that both events occurred by the probability that the first event has occurred. The
formula is P B∣A =
Example 1. A box contains black chips and white chips. A person selects two chips without replacement. If the
probability of selecting a black chip and a white chip is , and the probability of selecting a black chip on the
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first draw is , find the probability of selecting the white chip on the second draw, given that the first chip
selected was black chip.
Solution: Let B= selecting a black chip
W= selecting a white chip
P W∣B =
=
=
Example 2. The probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the
probability that Sam cannot find a legal parking space and has to park in the parking zone is 0.20. On Tuesday,
Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking
ticket.
Solution: Let N=parking in a no-parking zone T=getting a ticket
.
P T∣N =
= . =0.30
PROBABILITIES FOR “AT LEAST”
Complementary Event Rule.
1.
2.
3.
4.
P(E)=1-P(̅ )
A game is played by drawing 4 cards from an ordinary deck and replacing each card after it
is drawn. Find the probability that at least 1 ace is drawn.
A coin is tossed 5 times. Find the probability of getting at least 1 tail?
3. The Neckware Association of America reported that 3% of ties sold in the United States
are bow ties. If 4 customers who purchased a tie are randomly selected, find the probability
that at least 1 purchased a bow tie.
4. In a shooting game, the probability that Kim, Ken, and Karla can hit the target is 1/3, ¼,
and 1/6, respectively. What is the probability that the target will be hit if they all shoot at it
once?
FUNDAMENTAL PRINCIPLE OF COUNTING: MULTIPLICATION RULE
In sequence of events in which the first one has m1 possibilities, the second has m2, the third has m3
and so on and the total number of possible outcomes will be
m1•m2•m3•...mn
where n is the number of events.
Examples:
1.
2.
3.
4.
In a restaurant, a person can choose from the 8 viands, plain, garlic or java rice, 5 kinds of
beverages and 6 kinds of desserts. In how many ways can this person choose what to have if
he is to order one from each group?
In how many ways can 4 boys and 3 girls be seated in a row of 7 seats if the end seats are to
be occupied by boys?
In how many ways can 3 men be assigned consecutive seats in a row of 7 seats?
In how many ways can 4 boys and 3 girls be seated in a row of 7 seats with the girls in
consecutive seats?
FUNDAMENTAL PRINCIPLE OF COUNTING: ADDITION RULE
In sequence of events in which the first on has m1 possibilities, the second has m2, the third has m3, and so on,
and if the events are mutually exclusive, then the total number of possible outcomes will be
m1+m2+m3+...mn
where n is the number of events.
1.
2.
3.
How many numbers greater than 5 000 can be formed the digits 1, 2, 3, 4, and 5 using each
digit only once in each number?
How many numbers greater than 5000 of four different digits each, can be formed by the use
of the digits 0, 2, 3, 4, 8, 9?
How many integers between 1000 and 9000 may be formed from the digits 2, 4, 5, 6 and 7?
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Permutations
Factorials are used extensively in mathematics.
Lek k be a positive integer. Then the product of the first k positive integers is called k factorial and is denoted
by the symbol k! . Thus,
k!=k• k-1 • k-2 •...3•2•1
Note: 0!=1 and 1!=1
For example. 5!= 5•4•3•2•1=120
Every arrangement in order of a set of things is called permutations. It refers to a set of objects is any
arrangement of the said objects in definite order. Thus, the set of the letters m, s, a, if we use all of them, can
be arranged in the following orders:
msa
mas
ams
sma
sam
asm
Cases to be followed:
1. Permutations of Objects Taken all Together (n!)
Ex. In how many specific ways can three books, Statistics, Algebra and Biology, be arranged on a
shelf?
2. Permutations of n objects taken r at a time.
a.
b.
!
=
!
How many 5-letter words can be formed from the word FORMALITY?
How many four-letter permutations can be formed from the letters in the word “heptagon”?
c. A school musical director can select 2 musical plays to present next year. One will be presented in
the fall, and one will be presented in the spring. If she has 9 to pick from, how many different
possibilities are there?
d. The advertising director for a television show has 7 ads to use on the program. If she selects 1 of
them for the opening of the show, 1 for the middle of the show, and 1 for the ending of the show,
how many possible ways can this be accomplished?
3. Permutation of n objects Not all Distinct.
=
!
! ! !.. !
Ex. How many permutations can be made with the word CONCOCTION?
Ex. In how many ways can you arrange the letters of the word “PROBABILITY”?
Ex. In how many ways can you arrange the letters of the word “STATISTICS”?
4. Circular Permutations (n-1)!
Ex. In how many ways can eight guests be seated in a round table with eight chairs?
Ex. How many ways can you sit 10 people in a round table with 10 seats?
Combinations
Combination refers to a selection of objects with no attention given to their order of arrangement. Thus, msa,
mas, ams are all combinations, although they are different permutations.
1. Combination of n objects taken r at a time.
=
!
!
!
Ex. In how many ways can a reader select 3 books without regard to order from a set of 4 books?
Ex. From a deck of 52 cards, in how many ways can a hand of 13 cards be selected?
2. Combination in a series. nC1+nC2+nC3+...nCn=2n-1
Ex. In how many ways can a teacher assign at most six of her students to do a project?
Ex. How many committees can be formed from 5 people, if the committees consist of 1, 2, 3, 4 or 5
members
3. Combination of n objects taken all at the same time.
=
!
!
=
Ex. Evaluate 5C5.
Ex. In how many ways can seven members form a committee of 7?
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Probability and Counting Rules
1.
2.
3.
A box has contains 16 marbles, 7 of which are blue, 4 are red, and 5 are yellow. What is the
probability of drawing at random 3 marbles that are (a) blue; (b) yellow; (c) red; (d) 1 red and 2
yellow; (e) at least 1 blue; and (f) the marbles in order are blue, red and yellow?
Out of 5 men and 7 women, a committee consisting of 2 men and 3 women is to be formed. In how
many ways can this be done if
a. Any men and any women can be included
b. One particular women must be included on the committee
c. Two particular men cannot be included on the committee
A poker hand consists of five cards dealt from an ordinary deck of 52 playing cards.
a. How many possible poker hands are there?
b. How many different hands are there consisting of three aces and two kings?
c. How many different hands are there consisting of all red cards?
d. How many different hands are there consisting of 2 hearts, 2 diamonds and 1 spade?
4. A combination lock consists of the 26 letters of the alphabet. If a 3-letter combination is needed, find
the probability that the combination will consist of the letters ABC in that order. The same letter can
be used more than once. (Note: A combination lock is really a permutation lock.)
5. There are 8 married couples in a tennis club. If 1 man and 1 woman are selected at random to plan the
summer tournament, find the probability that they are married to each other.
LET’S DO AN EXERCISE
1.
2.
3.
4.
5.
6.
7.
8.
9.
In a class, there are 15 boys and 10 girls. Three students are selected at random. What is the
probability that 1 girl and 2 boys are selected?
From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the
cards being kings?
From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at
least 3 men are there on the committee. In how many ways can it be done?
A class photograph has to be taken. The front row consists of 6 girls who are sitting. 20 boys are
standing behind. The two corner positions are reserved for the 2 tallest boys. In how many ways can
the students be arranged?
In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the
vowels always come together?
A teacher is making a multiple choice quiz. She wants to give each student the same questions, but
have each student's questions appear in a different order. If there are twenty-seven students in the
class, what is the least number of questions the quiz must contain?
Find the probability that in a random arrangement of the letters of the word 'UNIVERSITY' the two I's
come together.
A special lottery is to be held to select a student who will live in the only deluxe room in a hostel.
There are 100 Year-III, 150 Year-II and 200 Year-I students who applied. Each Year-III's name is
placed in the lottery 3 times; each Year-II's name, 2 times and Year-I's name, 1 time. What is the
probability that a Year-III's name will be chosen?
How many words can be formed by re-arranging the letters of the word ASCENT such that A and T
occupy the first and last position respectively?
HYPERGEOMETRIC DISTRIBUTION
Definition
A set of N objects contains K objects classified as successes and N-K objects classified as failures. A
sample of size n objects is selected randomly (without replacement) from the N objects, where K N and
n
.
Let the random variables X denotes the number of successes in the sample. Then X is a
hypergeometric random variable and
=
x=max{0,
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( )(
)
( )
}
min{ , }
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The expression { , } is used in the definition of the range of X because the maximum number of
successes that can occur in the sample is the smaller of the sample size, n, and the number of successes
available, K.
Also, if n+K>N, at least n+K-N successes must occur in the sample. It should be noted that the
!
equation ( ) = !
is the number of a parts taken b at a time. The hypergeometric distribution is the
!
appropriate possibility model for sampling without replacement.
MEAN AND VARIANCE OF A HYPERGEOMETRIC DISTRIBUTION
If X is a hypergeometric random variable with parameters N, K, and n, then the mean
=
=
and the variance
=
=
1
Where
(
1
)
=
Here, p is interpreted as the proportion of successes in the set of N objects. For a hypergeoemetric random
variable, E(X) is similar to the mean of a binomial random variable. Also, V(X) differs from the result for a
binomial random variable only by the term shown below:
The term in the variance of a hypergeoemetric random variable
is called the finite population
correlation factor.
Examples:
1. Given that X has a hypergeoemetric distribution with N=100, n=4 and K=20. Determine the
following:
a. P(X=1)
b. P(X=6)
c. P(X=4)
d. the mean and variance of X.
2. A lot of 75 gaskets contains five in which the variability in thickness around the circumference of the
gasket is unacceptable. A sample of 10 gaskets is selected is selected at random, without
replacement. What is the probability that
a. none of the unacceptable gaskets is in the sample
b. at least one unacceptable gasket is in the sample
c. exactly one unacceptable gasket is in the sample
d. the mean number of unacceptable gaskets in the sample.
3. Of 50 manufactured steel rods in a production process by a company, 12 have defects. If 10 steel rods
are selected at random for inspection.
a. find the probability that exactly 3 of the 10 have defects
b. find the mean and variance of X.
4. A large bin contains 80 balls of which 32 are red balls and 48 are blue balls. Suppose 15 balls are
picked at random. Find the probability of getting 4 red balls, the mean number of red balls, and the
standard deviation of the number of red balls if the sample is picked
a. with replacement
b. without replacement.
5. Suppose a box contains five red balls and ten blue balls. If seven balls are selected at random without
replacement, find the probability that at least 3 red balls will be obtained.
THE BINOMIAL DISTRIBUTION
The binomial distribution is an example of a particular type of discrete probability distribution. It
may be referred to as a Bernoulli distribution, and the trials conducted are known as Bernoulli trials. They
were named in honour of the Swiss mathematician Jakob Bernoulli (1654-1705).
Bernoulli trials and sequences
A Bernoulli trial is an experiment in which the outcome is either a success or a failure. A Bernoulli sequence
is a sequence of Bernoulli trials in which:
1. the probability of each possible outcome is independent of the results of the previous trial; and
2. the probability of each possible outcome is the same for each trial.
Note: In a Bernoulli sequence, the number of successes follows the binomial distribution.
If X represents a random variable that has a binomial distribution, then it can be expressed as:
X~Bi(n,p) or X~B(n,p). It can be translated into words, X~Bi(n,p) means that X follows a binomial
distribution with parameters n (the number of trials) and p (probability of success).
Consider the experiment where a fair die is rolled four times. If X represents the number of times a 3
appears uppermost, then X is a binomial variable. Obtaining a 3 will represent a success and all other values
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will represent a failure. The die is rolled four times so the number of trials, n, equals 4 and the probability, p,
of obtaining a 3 is equal to 1/6. Using the shorthand notation, X~Bi(n,p) becomes X~Bi(4, 1/6).
The procedure for determining the individual probabilities can become tedious, particularly once the
number of trials increases. Hence if X is a binomial random variable, its probability is defined as follows.
Pr(X=x)=
where x= 0, 1, 2, … n. That is: x= the occurrence of the successful outcome.
Pr(X=x)=
1
where x= 0, 1, 2, … n. Here, the probability of failure, q, is replaced by 1p.
Note: It should be expected that the sum of the probabilities is 1.
The parameters n and p affect the binomial probability distribution curve as follows:
a. If p<0.5, the graph is positively skewed.
b. If p=0.5, the graph is symmetrical or is a normal distribution curve.
c. If p>0.5, the graph is negatively skewed.
d. When n is large and p=0.5, the interval between the vertical columns decreases and the graph
approximates a smooth hump or bell shape.
Example 1. Determine which of the following sequences can be defined as Bernoulli sequences.
a.
b.
c.
Rolling an 8-sided die numbered 1 to 8 forty times and recording the number of 6s obtained.
Drawing a card from a fair deck with replacement and recording the number of aces.
Rolling a die 60 times and recording the number that is obtained.
Example 2. A binomial variable, X, has the probability function Pr (X=x) = 6
0.4
0.6
x=0,
1,…6. Find:
a. n, the number of trials
b. p, the probability of success
c. the probability distribution for x as a table.
Example 3. A fair dice is rolled five times. Find the probability of obtaining:
a.
b.
exactly four 5s
exactly two even numbers
Example 4. A new drug for hay fever is known to be successful in 40% of cases. Ten hay fever sufferers take
part in the testing of the drug. Find the probability that:
a. four people are cured
b. no people are cured
c. all 10 are cured.
Example 5. Find Pr(X 3 if X has a binomial distribution with the probability of success, p, and the number of
trials, n, given by p=0.3, n=5.
Example 6. A bag contains 4 red and 3 blue marbles. A marble is selected at random and replaced. The
experiment is repeated 7 times. Find the probability that:
a. all 7 selections are red
b. at least 5 are red
c. no more than 2 are red.
Example 7. Seventy per cent of all scheduled trains through Westbourne station arrive on time. If 10 trains go
through the station every day, find:
a. the probability that at least 8 trains are on time
The Multinomial Distribution
Recall that in order for an experiment to be binomial; two outcomes are required for each trial.
But if each trial in an experiment has more than two outcomes, a distribution called the multinomial
distribution must be used. For example, a survey might require the responses of “approve,”
“disapprove,” or “no opinion.” In another situation, a person may have a choice of one of five activities
for Friday night, such as a movie, dinner, baseball game, play, or party. Since these situations have more
than two possible outcomes for each trial, the binomial distribution cannot be used to compute
probabilities.
The multinomial distribution can be used for such situations if the probabilities for each trial
remain constant and the outcomes are independent for a fixed number of trials.
The events must also be mutually exclusive.
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Note: Again, note that the multinomial distribution can be used even though replacement is not done, provided
that the sample is small in comparison with the population.
Examples:
1. In a large city, 50% of the people choose a movie, 30% choose dinner and a play, and 20% choose
shopping as a leisure activity. If a sample of 5 people is randomly selected, find the probability that 3
are planning to go to a movie, 1 to a play, and 1 to a shopping mall.
2. A small airport coffee shop manager found that the probabilities a customer buys 0, 1, 2, or 3 cups of
coffee are 0.3, 0.5, 0.15, and 0.05, respectively. If 8 customers enter the shop, find the probability that
2 will purchase something other than coffee, 4 will purchase 1 cup of coffee, 1 will purchase 2 cups,
and 1 will purchase 3 cups.
3. A box contains 4 white balls, 3 red balls, and 3 blue balls. A ball is selected at random, and its color is
written down. It is replaced each time. Find the probability that if 5 balls are selected, 2 are white, 2
are red, and 1 is blue.
POISSON DISTRIBUTION
The Poisson distribution, named after the French mathematician Simeon D. Poisson in 1837 is
another important probability distribution of a discrete random variable that has many applications. The
Poisson distributions is applied to experiments with random and independent occurrences.
Given an interval of real numbers, assume events occur at random throughout the interval. If the
interval can be partitioned into subintervals of small enough length such that
1. the probability of more than one event in an subinterval is zero.
2. the probability of one event in a subinterval is the same for all subinterval is the same for all
subintervals and proportional to the length of the subinterval, and
3. the event in each subinterval is independent of other subintervals, the random experiment is called a
Poissson process.
Conditions to apply Poisson probability distribution
1. x is discrete random variable
2. the occurrences are random
3. the occurrences are independent
Some of the phenomena that follow the Poisson distribution are:
1. counts of flaws in castings
2. the number of vehicles on a highway
3. the number of costumers visiting a bank
4. the number of accidents that occur on a given highway during a period of time
5. the number of telephone calls
6. counts of power outages
7. counts of atomic particles emitted from a specimen
Poisson distribution Formula
P(x;
=
!
;
x=0, 1, 2,..
where
e is the constant 2.7183 used in connection with natural logarithm
is the average (mean) of the distribution
x is the specific value in which we are interested
If x is a Poisson random variable with parameter , then
the mean
=
= , and
the variance
=
=
Examples
1. In statistics course final examination, 15% of the students fail. Find the probability that in random of
100 students in that statistics class who took the final examination exactly 20 will fail. Use the
Poisson formula.
2. Suppose an average of 5 calls for service per hour is received by a machine repair office. What is the
probability that exactly two calls for service will be received in a randomly selected hour?
3. Determine the probability that
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4.
5.
6.
7.
a. two people out of 500 have a birthday on New Year’s day.
b. At least one person out of 500 has a birthday on New Year’s day?
Surveys found that 1.5% of occupied units have 7 or more people living within. Use the Poisson
distribution to determine the approximate probability that, of 200 randomly selected occupied
housing units, there are
a. none with 7 or more persons
b. 3 or more with 7 or more persons
A survey found that the traffic flowing through an intersection with an average of 3 cars per 30
seconds. Assume the traffic flow can be modelled as a Poisson distribution.
a. find the probability of no cars through the intersection within 30 seconds
b. find the probability of 3 or more cars through the intersection within 30 seconds.
A student’s campus newspaper in a particular university contains an average of 1.2 typographical
errors per page. Find the probability that a randomly selected page of this newspaper will contain
exactly 4 typographical errors using the Poisson formula.
A hardware store in a big city receives an average of 9.8 telephone calls per hour. Find the probability
that exactly 6 telephone will be received at this store during a certain hour.
LET’S DO AN EXERCISE
1.
2.
3.
4.
5.
6.
In the manufacture of glassware, bubbles can occur in the glass which reduces the status of the
glassware to that of a ‘second’. If, on average, one in every 1000 items produced have a bubble,
calculate the probability that exactly six items in a batch of three thousand are seconds.
An automatic camera records the number of cars running a red light at an intersection (that is, the
cars were going through when the red light was against the car). Analysis of the data shows that on
average 15% of light changes record a car running a red light. Assume that the data has a binomial
distribution. What is the probability that in 20 light changes there will be exactly three (3) cars
running a red light?
A Statistics department purchased 24 hand calculators from a dealer in order to have a supply on
hand for tests for use by students who forget to bring their own. Although the department was not
aware of this, five of the calculators were defective and gave incorrect answers to calculations. When
a test is being written, students who have forgotten their own calculators are allowed to select one of
the Department's (at random). Suppose at the first test of the term, four students forgot to bring their
calculators. What is the probability that exactly one of these students selects a defective calculator?
A graduate statistics course has seven male and three female students. The professor wants to select
two students at random to help her conduct a research project. What is the probability that the two
students chosen are female?
In a school survey 68% of the students have an Android device. What is the probability that 12 of a
selecting of 20 have Android devices?
The average number of homes sold by the Acme Realty Company is 2 homes per day. What is the
probability that exactly 3 homes will be sold tomorrow?
DISCRETE RANDOM VARIABLES
Random variables are expressed as capital letters, usually from the end of the alphabet ( for example
X, Y, Z) and the value they can take on is represented by lowercase letters ( for example x, y, z respectively)
Random variable is categorized into two, discrete and continuous random variable. Discrete random
variables generally deal with number or size or outcomes were able to be counted. Continuous random
variables generally deal with quantities which can be measured, such as mass, height or time.
Discrete random variables generally deal with number or size and are able to be counted. A discrete
probability distribution exists only if the following two characteristics are satisfied.
a. Each probability lies in a restricted interval 0 Pr =
1.
b. The probabilities of a particular experiment sum to 1, that is, ∑ Pr X=x =1.
If these two characteristics are not satisfied, then there is no discrete probability distribution.
Example 1. Which of the following represent discrete random variables?
a.
The number of goals scored at a football match.
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b.
c.
d.
e.
The height of students in a Maths Methods class.
Shoe sizes.
The number of girls in a five-child family.
The time taken to run a distance of 10 kilometres in minutes.
DISCRETE PROBABILITY DISTRIBUTIONS
Example 2. Let X represent the number of tails obtained in three tosses. Draw up a table which displays the
values the discrete random variable random variable can assume and the corresponding probabilities.
Example 3. Which of the following tables represent a discrete probability distribution?
X
0
1
2
3
Pr
(X=x)
0.2
0.5
0.2
0.1
X
-2
0
5
7
Pr
(X=x)
-0.2
0.3
0.5
0.4
X
Pr (X=x)
0
0.5
2
0.3
4
0.1
6
0.1
X
-1
0
1
2
Pr
(X=x)
0.2
0.1
0.3
0.3
Example 4. Find the value of k for each of the following discrete probability distributions.
1.
X
Pr (X=x)
2.
X
Pr (X=x)
1
0.2
3
k
5
0.2
7
0.3
9
0.1
0
5k
1
6k
2
4k
3
3k
4
2k
Example 5. Show that the function
=
Example 6. Show that the function
=
5
3 , where x= 0, 1, 2, 3 is a probability function.
6
, where x=2, 3, 4, 5 is a probability function.
EXPECTED VALUE OF DISCRETE RANDOM DISTRIBUTIONS
The expected value of a discrete random variable, X, is the average value of X. It is also referred to as the mean
of X or the expectation.
 The expected value of a discrete random variable, X, is denoted by E X or the symbol mu . It is
defined as the sum of each value of X multiplied by its respective probability; that is,
E(X)= x1Pr(X=x1) + x2Pr(X=x2) + x3Pr(X=x3) +... + xnPr(X=xn)= ∑
=
 The expected value of a discrete random variable, X, is defined by the rule
=∑
= .
∑
Also
=
Pr = .
 A game is considered fair if the cost to play the game is equal to the expected gain.
 A fair game is one in which
= 0.
 The expected value of a linear function can be calculated using the expectation theorems:
=
=
=
=

Note: The expected value will not always assume a discrete value.
Example 7. Find the expected value of a random variable which has the following probability distribution.
x
Pr(X=x)
1
2/5
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1/10
3
3/10
4
1/10
5
1/10
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Example 8. Find the unknown probability, a, and hence determine the expected value of a random variable
which has the following probability distribution.
X
2
Pr(X=x) 0.2
4
0.4
6
a
8
0.1
10
0.1
Example 9. Find the values of a and b of the following probability distribution if
x
Pr (X=x)
1
0.1
2
0.1
3
A
4
0.3
5
0.2
6
b
= 4.2
7
0.2
Example 10. Niki and Melanie devise a gambling game based on tossing three coins simultaneously. If three
heads or three tails are obtained, the player wins $20. Otherwise the player loses $5. In order to make a profit
they charge each person two dollars to play.
a. What is the expected gain to the player?
b. Do Nikki and Melanie make a profit?
c. Is this a fair game?
Example 11. A random variable has the following probability distribution.
X
Pr (X=x)
Find:
a)
c)
2
4
1
2
0.25 0.26
b) 3
d)
3
0.14
4
0.35
VARIANCE AND STANDARD DEVIATION OF DISCRETE RANDOM DISTRIBUTIONS
Variance is an important feature of probability distributions as it provides information about the
spread of the distribution with respect to the mean. If the variance is large, it implies that the possible values
are spread (or deviate) quite a distance from the mean. A small variance implies that the possible values are
close to the mean. Variance is also called a measure of spread or dispersion.
The variance is written as ar X and denoted by the symbol
(sigma squared). It is defined as the
expected value (or average) of the squares of the spreads (deviations) from the mean.

The variance, Var(X) or
is defined by the rule:
Var (x)=


The variance of a linear function can also be calculated by the following rule:
Var
=
The standard deviation, SD X or σ, is defined by the rule:
SD(X)=√
=√
Example 12. Find the expected value and variance of the following probability distribution table.
x
1
2
3
4
5
Pr (X=x) 0.15 0.12 0.24 0.37 0.12
Example 13. Find the variance of 2Y+1 for the following probability distribution table.
Y
Pr (Y=y)
0
0.25
1
0.35
2
0.2
3
0.2
Example 14. X is a discrete random variable with the following probability distribution.
X
3
4
6
K
Pr(X=x) 0.15 0.3 0.45 0.1
Find the value of k, a positive integer, if the variance is 1.7475.
Example 15. A random variable has the following probability distribution.
X
Pr(X=x)
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¼
1
3/8
2
1/8
3
¼
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Calculate the expected value, the variance and the standard deviation.
Example 16. In order to encourage car pooling, a new toll is to be introduced on the Eastgate Bridge. If the car
has no passengers, a toll of $2 applies. Cars with one passenger pay a $1.50 toll, cars with two passengers pay
a $1 toll and cars with 3 or more passengers pay no toll. Long-term statistics show that the number of
passengers (X) follows the probability distribution given below.
x( no. of passengers) 0
1
Pr(X=x)
0.4 0.35
a. Construct a probability distribution of the toll paid.
b. Find the mean toll paid per car.
c. Find the standard deviation of tolls paid.
2
0.2
3
0.05
EXPECTED VALUE, VARIANCE AND STANDARD DEVIATION OF THE BINOMIAL DISTRIBUTIONS
If X is a random variable and X~Bi(n,p) then: E(X)=np, Var(X)=npq and SD(X)= √
distribution must be binomial for these rules to apply.
. Note: the
Example 24: If the random variables, X, is such that X~Bi(8, 0.3) and has the following probability
distribution. Find the expected value, variance and standard deviation.
X
Pr (X=x)
0
0.05765
1
0.19765
2
0.29648
3
0.25412
4
0.13614
5
0.04668
6
0.01000
7
0.00122
8
0.00007
Example 25. The random variable X follows a binomial distribution such that X~Bi(40, 0.25). Determine the:
a.
expected value
b. variance and standard deviation
Example 26. A fair die is rolled 15 times. Find:
a.
the expected number of 3s rolled b. the probability of obtaining more than the expected number of
3s.
Example 27. A binomial random variable has an expected value of 14.4 and a variance of 8.64. Find:
a.
the probability of success, p
b. the number of trials, n.
Example 28. A new test designed to assess the reading ability of students entering high school showed that
10% of the students displayed a reading level that was inadequate to cope with high school. If 400 students
are selected at random, find the expected number of students whose reading level is inadequate to cope with
high school.
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