1. Lecture Week 3:
Probability, Probability
Distribution & Sampling
Distribution
Prepared by:
Dr. Nurul Syaza Abdul Latif
2. Probability
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Probability is a likelihood or chance that an event will occur.
Example: The chance of picking a red ball from a box
containing red and blue balls, or the chance of raining today
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Definition: If S is the sample space (a set containing all the
possibility outcomes of an experiment) and A is an event (a
subset of S) then the probability of A is
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4. Example 1
Consider of an experiment in tossing a die.
There are six possible outcomes when a die
is tossed once.
Sample space S={1,2,3,4,5,6}
Suppose A is the event of getting odd
number i.e. A={1,3,5}. Therefore n(A)=3
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Hence, the probability of getting of getting
an odd number is 0.5
5. Venn Diagram is
occasionally helpful in
determining the probability
of simple events
6. Concepts: Combinations &
Permutations
Definition (Combinations): If there are N elements
in a population and we want a sample of size r, then
the number of ways of selecting r elements from a
total of N is given by
7. In your plant disease management class,
you are assigned to read any 4 books
from a list of 10 books. How many
different groups of 4 are available from
the list of 10?
8. Definition (Permutations): If there are N elements
and we want to select r elements but the order in
which the elements are selected is IMPORTANT,
then the number of ways this can be done is
9. Example 3: The number of possible ordered
seating arrangements for eight people in five
chairs.
10. Example 4:
In a lucky draw box contains brand new iPhone 5c. There
are 5 yellow, 4 red and 3 blue in that box. In how many
ways can Ali choose 4 iPhones from the box? What is the
probability Ali chooses 2 yellow iPhones, one red iPhone
and one blue iPhone?
12. Probability Rules
P(entire sample space)=1
For any events A: 0< P(A) <1
if A’ is the compliment of A, then P(A’)=1-P(A)
Events A & B are independent if P(A)=P(A|B)
Multiplication rules
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Conditional probability:
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Events A & B mutually exclusive if
Addition rules
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13. Example 5:
Suppose 20 students share the same
floor of a dormitory. Eleven of them
took Statistics class, eight took
Chemistry class and three took both
Statistics and Chemistry classes. A
student is chosen at random from the
20 students. What is the probability he
took either the Statistics or Chemistry
classes? What is the probability he did
not take any of two classes?
22. 2. Probability Distributions
2.1 Binomial Distributions
The experiment involves n identical trials or
observations.
Each trial or observation has only two possible,
mutually exclusive outcomes denoted as success
or as failure. The outcome of interest to the
researcher is labelled a success.
Each trial is independent of the previous trial
The probability of getting success is p in any one
trial, and the probability of getting failure is q=1-
p in any one trial. The terms p and q remain
constant throughout the experiment.
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24. Example 9: Given that r ~ b(10,0.2). Find:
P(r=0)
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P(r=2)
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P(r<=2)
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the mean
the standard deviation
25. Example 10:
Privacy is a concern for many users of the
internet. One survey showed that 59% of
internet users are somewhat concerned about
the confidentially of their email. Based on
this information, what is the probability that
a random sample of 10 internet users, 6 are
concerned about the privacy of their email?
27. Example 11:
A biologist is studying a new hybrid of tomato.
It is known that the seeds of the hybrid tomato
have probability of 0.70 germinating. The
biologist plants 6 seeds.
a)What is the probability that EXACTLY four
seeds will germinate?
b)What is the probability that AT LEAST four
will germinate?
c)What is the mean and
standard deviation?
29. Solution Example 11
b) AT LEAST four will germinate?
c) What is the mean and standard deviation?
30. 2.2 Normal Distributions
The normal probability distribution is a
continuous probability distribution whose
density function has a bell-shaped graph
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The distribution is both symmetrical and
mesokurtic. Symmetrical means that each half
of the distribution is a mirror image of the
other half. Mesokurtic means that it is neither
flat nor peaked in terms of the distribution of
observed values.
31. The normal probability distribution is important in statistical inference for the
following reasons:
The measurements obtained in many random processes are known to
follow this distribution
The normal probability distribution fits many human characteristics
such as height, speed, IQ, scholastic achievement and years of life
expectancy. Living things in nature such as trees, animals, and insects
also have many characteristics that are normally distributed.
The area under the curve in figure yields the probabilities, so the total
area under the curve is 1.
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A probability value for a continuous random variable can be
determined only for an interval of values. That is
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33. Standard Normal Distribution
Because the function is so complex, using it to determine areas under the curve
is difficult and time-consuming. Thus table of normal probabilities can be used
to analyse normal distribution problems. This is defined by standard normal
distribution denoted by z. Standard normal distribution is the normal
probability distribution with
38. 3. Sampling Distribution
A population consists of the totality of the observations with which we are
concerned. For example: the heights of UMK students, the length of crop
trees in a plantation area etc. A sample is a subset of a population.
The main purpose in selecting random samples is to get information about
the unknown population parameters such as the population mean and the
population variance.
e.g. : Suppose we want to measure the heights of every students in UMK. It
would be quite impossible to measure the height of every students in UMK
in order to compute the value , representing the population mean.
Instead of a large random sample (e.g. specific to only campus jeli) is
selected and the mean height for the sample is obtained. The value
is now used to make an inference concerning the true mean height .
Since many random samples are possible from the same population, we
would expect to vary from sample to sample. That is is a value of a
random variable that we represent by x . Such random variable is called
statistics.
Since a statistics is a random variable that depends only on the observed
sample, it must have a probability distribution. The probability distribution
of a statistic is called a sampling distribution.
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40.
41. a) probability that a single trout taken at random
from the pond is between 8 and 12 inches long?
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42. a) probability that the mean length x ̄of five trout
taken at random is between 8 and 12 inches ?
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