Unit 1
Review of Probability and basic statistics
Classical Probability
• Classical probability is the statistical concept that
measures the likelihood (probability) of something
happening.
• In a classic sense, it means that every statistical
experiment will contain elements that are equally likely
to happen (equal chances of occurrence of something).
Therefore, the concept of classical probability is the
simplest form of probability that has equal odds of
something happening.
Classical Probability
For example:
• Rolling a fair die. It’s equally likely you would get a 1, 2, 3, 4, 5, or 6.
• Selecting bingo balls. Each numbered ball has an equal chance of
being chosen.
• Guessing a multiple choice quiz (MCQs) test with (say) four possible
answers A, B, C or D. Each option (choice) has the same odds (equal
chances) of being picked (assuming you pick randomly and do not
follow any pattern).
• Formula for Classical Probability.
The probability of a simple event happening is the number of times
the event can happen, divided by the number of possible events.
P(A) = f / N.
Classical Probability
• Dividing the number of events by the number of
possible events is very simplistic, and it isn’t suited to
finding probabilities for a lot of situations.
• For example, natural events like weights, heights, and
test scores need normal distribution probability charts
to calculate probabilities. In fact, most “real life”
things aren’t simple events like coins, cards, or dice.
You’ll need something more complicated than
classical probability theory to solve them.
Subjective Probability
• Subjective Probability is based on your beliefs.
• For example, you might “feel” a lucky streak coming on.
• When logic and past history are not appropriate probability
values can be assessed subjectively.
• The accuracy of subjective probability depends on the
experience and judgment of the person making the estimates.
• There are several methods for making subjective probability
assessments.
• Opinion polls can be used to help in determining subjective
probabilities for possible election returns and potential
political candidates.
Empirical probability
• Empirical probability is based on experiments. You
physically perform experiments and calculate the
odds from your results.
• The empirical probability, relative frequency, or
experimental probability of an event is the ratio of the
number of outcomes in which a specified event
occurs to the total number of trials, not in a
theoretical sample space but in an actual experiment.
In a more general sense, empirical probability
estimates probabilities from experience and
observation.
Empirical probability
• An advantage of estimating probabilities using
empirical probabilities is that this procedure is
relatively free of assumptions.
• A disadvantage in using empirical probabilities
arises in estimating probabilities which are
either very close to zero, or very close to one. In
these cases very large sample sizes would be
needed in order to estimate such probabilities to
a good standard of relative accuracy.
Axiomatic Probability
• Axiomatic Probability is a type of probability that has
a set of axioms (rules) attached to it.
• For example, you could have a rule that the
probability must be greater than 0%, that one event
must happen, and that one event cannot happen if
another event happens.
Fundamental concepts
There are 2 basic rules regarding the mathematics of
probability:
1. The probability P , of an event or state of nature
occurring is greater than or equal to 0 and less than
or equal to 1. that is 0 ≤ P (event) ≤ 1.
2. The sum of the simple probabilities for all possible
outcomes of an activity must equal 1.
Mutually Exclusive Events
• Two events are mutually exclusive or disjoint if both
of them cannot occur at the same time.
• The set of outcomes of a single coin toss, which can
result in either heads or tails , but not both.
• We can’t run backwards and forwards at the same
time. The events “ running forward” and “ running
backwards” are mutually exclusive.
Collectively Exhaustive events
• collectively exhaustive events is that their union must
cover all the events within the entire sample space.
For example, events A and B are said to be
collectively exhaustive if
A ∪ B = S where S is the sample space.
Adding Mutually exclusive events
• The law of addition is simply as follows:
P( event A or event B) = P (event A) + P (event B)
Adding Not Mutually exclusive events
• When two events are not mutually exclusive,
P(event A or event B) = P (event A) + P (event B)
- P( event A and event B both occurring)
Statistically independent events
Events may be either independent or dependent.
When they are independent, the occurrence
of one event has no effect on the probability of
occurrence of the second event.
Statistically dependent events
• When events are statistically dependent, the
occurrence of one event affects the
probability of occurrence of some other event.
• The three types of probability under both statistical
independence and statistical dependence are (1)
marginal, (2) joint, and (3) conditional
• A marginal probability is the probability of an event
occurring.
• A joint probability is the product of marginal
probabilities. P(AB) = P(A)* P(B)
• A conditional probability is the probability of an event
occurring given that another event has
taken place. it is expressed as P(A│B) or “the probability of
event B, given that event A has occurred.”
• If independent events occurrence of one in no way affects the
outcome of another, P(A│B) = P(A)and
P(B│A) = P(B).
• conditional probability under dependence is somewhat more
involved than it is under independence.
• The formula for the conditional probability of A, given that
event B has taken place, is stated as P(A │ B)=P(AB)/P(B)
• the formula for a joint probability is
P(AB) = P(A │ B)P(B)
Q1.In the past 30 days, Roger’s Rural Roundup has sold
either 8, 9, 10, or 11 lottery tickets. It never sold
fewer than 8 or more than 11. Assuming that the past
is similar to the future, find the probabilities for the
number of tickets sold if sales were 8 tickets on 10
days, 9 tickets on 12 days, 10 tickets on 6 days, and
11 tickets on 2 days.
Solution:
SALES NO. DAYS PROBABILITY
8 10 0.333
9 12 0.4
10 6 0.2
11 2 0.067
Total 30 1.000
Q2.Your professor tells you that if you score an 85 or
better on your midterm exam, then you have a 90%
chance of getting an A for the course. You think you
have only a 50% chance of scoring 85 or better. Find
the probability that both your score is 85 or better and
you receive an A in the course.
Solution:
P(A and 85) = P(A │85) * P(85) = (0.90)(0.50)
= 45%
Q3.A statistics class was asked if it believed that all
tests on the Monday following the football game
win over their archrival should be postponed
automatically. The results were as follows:
Strongly agree 40,Agree 30,Neutral 20,Disagree
10,Strongly disagree 0.Transform this into a
numeric score, using the following random variable
scale, and find a probability distribution for the
results: Strongly agree 5,Agree 4,Neutral 3,Disagree
2,Strongly disagree 1
Compute the expected value, the variance and standard
deviation for the random variable X.
• Solution
OUTCOME PROBABILITY, P (X)
Strongly agree (5) 0.4
Agree (4) 0.3
Neutral (3) 0.2
Disagree (2) 0.1
Strongly disagree (1) 0.0
Total 1
E(X) = 4 , variance = 1,
standard deviation = 1
Bayes’ Theorem
• It is used to incorporate additional information as it is
made available and help create revised or posterior
probabilities.
• In simple words we can take new or recent data and
then revise and improve upon our old probability
estimates for an event.
Bayes’ Theorem
Bayes’ Theorem
Example 1
Example 1....
Example 2
Example 2....
Example 2....
Example 2.....
Random Variable
• A random Variable assigns a real number to
every possible outcome or event in an
experiment.
• Normally represented by X,Y....
• When the outcome itself is numerical or
quantitative, the outcome number can be the
random variable.
Example 3
• Consider refrigerator sales at an appliance
store. The number of refrigerators sold during
a given day can be the random variable.
X= number of refrigerators sold during the day
Example 4
Whenever the experiment has quantifiable outcomes,
it is beneficial to define these quantitative outcomes
as the random variable.
Example 5
• When the outcome itself is not numerical or
quantitative, it is necessary to define a
random variable that associates each outcome
with a unique real number.
Two types of Random variables
• A random variable is a discrete random
variable if it can assume only a finite or limited
set of values.
• A continuous random variable that has an
infinite or an unlimited set of values.
Probability distribution
• Three rules required for probability distribution
1. The events are mutually exclusive and collectively
exhaustive
2. The individual probability values are between 0 and 1
inclusive,
3. The total of the probability value sum to 1.
• Discrete probability distribution was computed using the
relative frequency .
• With a continuous probability distribution, there is a
continuous mathematical function that describe the
probability distribution. This function is called the probability
density function or simply the probability function.
Expected value and Variance of a Discrete
probability Distribution
• The graph of discrete probability distribution gives us a
picture of its shape. It helps us identify the central tendency of
the distribution, called the mean or expected value, and the
amount of variability or spread of the distribution, called the
variance.
Expected value and Variance of a Discrete
probability Distribution
• Variance of a probability distribution is a
number that reveals the overall spread or
dispersion of the distribution.
Probability distribution of a continuous random
variable
• With a continuous probability distribution, there is a
continuous mathematical function that describe the probability
distribution. This function is called the probability density
function or simply the probability function. the probability
function can be graphed , and the area underneath the curve
represents probability.
Example 6
Example 6....
Example 7
• Compute the expected value , variance and
standard deviation in example 6
Binomial Distribution
Binomial Distribution
Example 8
A candidate for public office has claimed that
60% of voters will vote for her. If 5 registered
voters were sampled, what is the probability
that exactly 3 would say they favour this
candidate?
Example 8....
Solution:
We use the binomial distribution with n= 5,p
=0.6 and r = 3;
P(exactly 3 successes in 5 trials) = 5C3 (0.6)3(0.4)2
= 0.3456
Example 9
Poisson distribution
• An important discrete probability distribution.
• It may be obtained as a limiting case of Binomial probability distribution
under following conditions.
n, the number of trials are indefinitely large i.e. n→∞
p, the constant probability of success for each trial is indefinitely small i.e.
p →0
n.p = m or λ (let) is finite
Example 10- 13
10) Comment on the following ,for a Poisson distribution, mean = 8
and variance = 7.
11) Between the hours 2 p.m. And 4 p.m. The average number of phone calls
per minute coming into the switch board of a company is 2.35 .find the
probability during one particular minute, there will be at most 2 phone
calls (given e-2.35 = 0.095374)
12) A manufacturer of toys knows that 5% of his product is defective. if he
sells toys in boxes of 100 and guarantees that not more than 10 toys will be
defective, what is the probability that a box will fail to meet the guaranteed
quality?
13) If a random variable X follows Poisson distribution such that P(X= 1) =
P(X=2), find the mean and variance of distribution and P(X=0).
Normal Distribution
• Each binomial distribution is defined by n, the number of trials
and p, the probability of success in any one trial.
• Each Poisson distribution is defined by its mean
• In the same way, each Normal distribution is identified by two
defining characteristics or parameters: its mean and standard
deviation.
• The Normal distribution has three distinguishing features:
It is unimodal, in other words there is a single peak.
It is symmetrical, one side is the mirror image of the other.
It is asymptotic, that is, it tails off very gradually on each side
but the line representing the distribution never quite meets the
horizontal axis
• It is symmetric around the point x = μ, which is at the same
time the mode, the median and the mean of the distribution.
• It is unimodal: its first derivative is positive for x < μ,
negative for x > μ, and zero only at x = μ.
Example 14
Solution14
Solution14
Solution14
Solution14
Q15. The length of the rods coming out of our new
cutting machine can be said to approximate a normal
distribution with a mean of 10 inches and a standard
deviation of 0.2 inch. Find the probability that a rod
selected randomly will have a length
(a) of less than 10.0 inches
(b) between 10.0 and 10.4 inches
(c) between 10.0 and 10.1 inches
(d) between 10.1 and 10.4 inches
(e) between 9.6 and 9.9 inches
(f) between 9.9 and 10.4 inches
(g) between 9.886 and 10.406 inches
• Solution 15:
First compute the standard normal distribution, the Z
value: z = (X-μ)/σ
Next, find the area under the curve for the given Z value
by using a standard normal distribution table.
(a) P(X < 10.0) = 0.50000
(b) P(10.0 < X < 10.4) = 0.97725 - 0.50000 = 0.47725
(c) P(10.0 < X < 10.1) = 0.69146 - 0.50000 = 0.19146
(d) P(10.1 < X < 10.4) = 0.97725 - 0.69146 = 0.28579
(e) P(9.6 < X < 9.9) = 0.97725 - 0.69146 = 0.28579
(f) P(9.9 < X < 10.4) = 0.19146 + 0.47725 = 0.66871
(g) P(9.886 < X < 10.406) = 0.47882 + 0.21566 = 0.69448
Q 16.Steve Goodman, production foreman for the
Florida Gold Fruit Company, estimates that the average
sale of oranges is 4,700 and the standard deviation
is 500 oranges. Sales follow a normal distribution.
(a) What is the probability that sales will be greater
than 5,500 oranges?
(b) What is the probability that sales will be greater
than 4,500 oranges?
(c) What is the probability that sales will be less
than 4,900 oranges?
(d) What is the probability that sales will be less
than 4,300 oranges?
Exponential Distribution
• The exponential distribution, also called the
negative exponential distribution, is used in
dealing with queuing problems.
• The exponential distribution often describes
the time required to service a customer.
• The exponential distribution is a continuous
distribution.
Exponential Distribution
• Its probability function is given by
Exponential Distribution
For the exponential distribution, the probabilities can be found
using the exponent key on a calculator with the formula
below. The probability that an exponentially distributed time
(X) required to serve a customer is less than or equal to time t
is given by the formula
The time period used in describing μ determines the units for
the time t. For example, if μ is the average number served per
hour, the time t must be given in hours. If μ is the average
number served per minute, the time t must be given in
minutes.
Using Excel for Expected value and Variance
Using Excel for Binomial Probability
• Excel has a (BINOMDIST) that will compute probabilities for
the binomial distribution.
n =the number of trials
p = the probability of a success on any single trial
r = the number of successes.
• Excel will provide the probability of exactly r successes with
the formula BINOMDIST (r, n, p, FALSE),FALSE indicates
that exactly r successes is desired.
• To find the probability that the number of successes is r or
less(a cumulative probability),we use BINOMDIST
(r, n, p, TRUE)
Q17. There are 10 questions on a true–false test. A student
feels unprepared for this test and randomly guesses
the answer for each of these.
(a) What is the probability that the student gets
exactly 7 correct?
(b) What is the probability that the student gets
exactly 8 correct?
(c) What is the probability that the student gets
exactly 9 correct?
(d) What is the probability that the student gets
exactly 10 correct?
(e) What is the probability that the student gets
more than 6 correct?
Solution 17
Q18. Suppose that a printer circuit board (PCB)
manufacturing company has been averaging
35 errors per month and that the company
has 2 PCB errors in a day. Arbitrarily choosing
a time unit of one day, the average number of
errors per day is 35/30 = 1.1667. the PCB
manufacturing company would like to know
( i) the probability of no error in a day (ii)
probability of 2 PCB errors in a day.
Solution 18:
Q19. If 10% of all disk drives produced on an
assembly line are defective, what is the
probability that there will be exactly one
defect in a random sample of 5 of these?
What is the probability that there will be no
defects in a random sample of 5?
• Solution 19 (i) 0.3114 (ii) 0.2119
REVISION
1.If only one event may occur on any one trial, then the
events are said to be
a. independent.
b. exhaustive.
c. mutually exclusive.
d. continuous.
2. A measure of central tendency is
a. expected value.
b. variance.
c. standard deviation.
d. all of the above.
3. To compute the variance, you need to know the
a. variable’s possible values.
b. expected value of the variable.
c. probability of each possible value of the variable.
d. all of the above.
4. The square root of the variance is the
a. expected value.
b. standard deviation.
c. area under the normal curve.
d. all of the above.
5.Which of the following is an example of a discrete distribution?
a. the normal distribution
b. the exponential distribution
c. the Poisson distribution
d. the Z distribution
6. The total area under the curve for any continuous distribution
must equal
a. 1.
b. 0.
c. 0.5.
d. none of the above.
7.Probabilities for all the possible values of a discrete random
variable
a. may be greater than 1.
b. may be negative on some occasions.
c. must sum to 1.
d. are represented by area underneath the curve.
8. If two events are mutually exclusive, then the probability of
the intersection of these two events will equal
a. 0.
b. 0.5.
c. 1.0.
d. cannot be determined without more information.
9. If P(A) = 0.4 and P(B) = 0.5and P(A and B) = 0.2,then P(A│B) =
a. 0.80.
b. 0.50.
c. 0.10
d. 0.40.
e. none of the above.
10. If P(A) = 0.4 and P(B) = 0.5and P(A and B) = 0.2 then P( A or
B) =
a. 0.7.
b. 0.9.
c. 1.1.
d. 0.2.
e. none of the above.
11.In a standard normal distribution, the mean is equal to
a. 1.
b. 0.
c. the variance.
d. the standard deviation.
12. The probability of two or more independent events
occurring is the
a. marginal probability.
b. simple probability.
c. conditional probability.
d. joint probability.
e. all of the above.
13.In the normal distribution, 95.45% of the population lies
within
a. 1 standard deviation of the mean.
b. 2 standard deviations of the mean.
c. 3 standard deviations of the mean.
d. 4 standard deviations of the mean.
14. If a normal distribution has a mean of 200 and a standard
deviation of 10, 99.7% of the population falls within what range
of values?
a. 170–230
b. 180–220
c. 190–210
d. 175–225
e. 170–220
15.If two events are mutually exclusive, then the probability of
the intersection of these two events will equal
a. 0.
b. 0.5.
c. 1.0.
d. cannot be determined without more information.
