1122

1. LET Review 3
Counting Techniques
Probability
Statistics

2. Counting
Techniques

3. Counting Techniques
Experiment
• any activity that can be done
repeatedly (e.g. tossing a coin,
rolling a die)
Sample space
• the set of all possible outcomes
in an experiment.
Sample point
• an element of the sample space

4. Counting Sample
Points
1. Fundamental Principle of
Counting (FPC)
• If a choice consists of k steps, of
which the first can be performed
in n1 ways, for each of these the
second can be performed in n2
ways, for each of these the third
can be performed in n3 ways....
and for each of these the kth can
be made in nk ways, then the
whole choice can be made in n1,
n2, n3, . . . nk ways.

5. Counting Sample
Points
2. Permutation
• Permutation is an arrangement of objects
wherein the order is important
a) Linear Permutation
Refers to n objects that are to be
arranged r objects at a time
𝑛𝑃𝑟=
𝑛!
𝑛 − 𝑟 !
, 𝑛 ≥ 𝑟
b) Circular Permutation
If n objects are to be arranged in a
circular manner
(𝑛 – 1)!
c) Permutations with Repetitions
The number of distinct permutations of n
things of which p are of one kind, q are of a
second kind,... r of a kth kind
𝑃 =
𝑛!
𝑝!𝑞!…𝑟!
, 𝑝 + 𝑞 + ⋯ + 𝑟 = 𝑛

6. Counting Sample
Points
3. Combination
• Combination is the arrangement of objects
regardless of order. In other words, the
order of arranging the objects is not
important.
𝑛𝐶𝑟=
𝑛!
𝑟! 𝑛 − 𝑟 !
, 𝑛 ≥ 𝑟

7. Probability

8. Probability
Probability
the likelihood of the occurrence
of an event
If E is any event, then .the
probability of an event denoted
by P(E) has a value between 0
and 1
If P(E) = 1, then E is sure to
happen.
If P(E) = 0, then E is impossible
to happen.

9. Probability
1. Theoretical Probability
Theoretically, the probability of
an event E, denoted by P(E), is
defined as
𝑃 𝐸 =
𝑛(𝐸)
𝑛(𝑆)
where
n(E) = number of favorable
outcomes
n(S) = number of possible
outcomes

10. Probability
2. Experimental Probability
The probability of an event may
also be obtained experimentally.
Suppose we want to find out the
probability of obtaining a tail in
a toss of coin.
We can perform an experiment
by tossing the coin 50 times and
record the number of
occurrences of tail. Suppose
that tail occurred 24 times, then
the probability of getting a tail
based on this experiment is
𝑃 𝑡𝑎𝑖𝑙 =
24
50

11. Statistics

12. Statistics
Statistics
the branch of mathematics used
to summarize quantities of data
and help investigators draw
sound conclusions
Sample
a specified set of measurements
or data, which is drawn from a
much larger body of
measurements or data called
the population

13. Statistics
Kinds of Sampling
1. Random sampling techniques
used to ensure that every
member, of the population has
an equal chance of being
included in the sample
representative of the entire
population
Two methods of random sampling
Lottery method
Use of the table of random
sampling

14. Statistics
Kinds of Sampling
2. Systematic sampling
technique which selects every
nth element of the population
for the sample
the starting point determined at
random from the first n
elements
3. Stratified random sampling
a technique of selecting simple
random samples from mutually
exclusive groupings or strata of
the population

15. Graphical
Representations of
Data
1. Histogram
A graphical picture of a
frequency distribution
consisting of a series of vertical
columns or rectangles, each
drawn with a base equal to
the class interval and a height
corresponding to the class
frequency.
The bars of a histogram are
joined together, that is, there
are no spaces between bars.

16. Graphical
Representations of
Data
2. Bar Chart
Uses rectangles or bars to
represent discrete classes of
data.
The length of each bar
corresponds to the frequency or
percentage of the given class or
category.

17. Graphical
Representations of
Data
3. Frequency Polygon
A special type of line graph,
where each class frequency is
plotted directly above the
midpoint or class mark of its
class interval and lines are then
drawn to connect the points.

18. Graphical
Representations of
Data
4. Pie Chart
An effective way of presenting
categorized (qualitative)
distributions, where a circle is
divided into sectors - pie-shaped
pieces - which are proportional
in size to the corresponding
frequencies or percentages.

19. Graphical
Representations of
Data
5. Pictogram
known as picture graph where
picture symbols are used to
represent values.

20. Graphical
Representations of
Data
MEASURES OF CENTRAL TENDENCY
A measure of central tendency is a
single, central value that
summarizes a set of numerical
data.
The measures of central tendency
are the mean, median and mode.
MEASURES OF VARIABILITY
A measure of variation or
variability describes how large
the differences between the
individuals are on a trait.
The common measures of
variability are range and standard
deviation.