3. Introduction to Counting and Probability
Some Terms
“Probability
is the branch of mathematics that
provide quantitative description of the likely
occurrence of an event.
Outcome – any possible result of an experiment or
operation
Sample space – the complete list of all possible
outcomes of an experiment or operation
Event – refers to any subset of a sample space
Counting – operation used to find the number of
possible outcomes
4. Introduction to Counting and Probability
Counting Problems
Counting problems are of the following kind:
“How
many combinations can I make with 5 Tshirts, 4 pairs of pants, and 3 kinds of shoes?
“How many ways are there to pick starting 5
players out of a 12-player basketball team?”
Most importantly, counting is the basis for
computing probabilities.
Example;:“What is the probability of winning the
lotto?”
5. Introduction to Counting and Probability
Counting Problems
Example:
Ang Carinderia ni Jay ay may breakfast promo
kung saan maaari kang makabuo ng combo meal
mula sa mga sumusunod:
SILOG
DRINKS
DESSERT
TAPSILOG
KAPE
SAGING
TOSILOG
MILO
BROWNIES
LONGSILOG
ICED TEA
BANGSILOG
6. Introduction to Counting and Probability
Counting Problems
Example:
Question: If you want to create a combo meal
by choose one of each kind, how many choices
can you have?
SILOG
DRINKS
DESSERT
TAPSILOG
KAPE
SAGING
TOSILOG
MILO
BROWNIES
LONGSILOG
ICED TEA
BANGSILOG
12. Fundamental Principle of Counting
The Product Rules
The Product Rule:
Suppose that a procedure can be
broken down into two successive
tasks. If there are n1 ways to do the
first task and n2 ways to do the second
task after the first task has been done,
then there are n1n2 ways to do the
procedure.
13. Introduction to Counting and Probability
The Product Rules
Generalized product rule:
If we have a procedure consisting of
sequential tasks T1, T2, …, Tn that can
be done in k1, k2, …, kn ways,
respectively, then there are n1 n2 …
nm ways to carry out the procedure.
14. Introduction to Counting and Probability
Basic Counting Principles
The two product rules are
collectively called the
FUNDAMENTAL PRINCIPLE
OF COUNTING
Woohoo…
15. Introduction to Counting and Probability
The Product Rules
Generalized product rule:
If we have a procedure consisting of sequential tasks T1, T2, …, Tm that
can be done in k1, k2, …, kn ways, respectively, then there are n1 n2 …
nm ways to carry out the procedure.
T1
k1
T2
k2
T3
k3
…
…
Tn
Tasks
kn
No. of
ways
no. of outcomes in all = k1k2k3 ...kn
16. Introduction to Counting and Probability
Basic Counting Principles
Example 1
If you have 5 T-shirts, 4 pairs of pants,
and 3 pairs of shoes, how many ways
can you choose to wear three of them?
Solution
The number of ways is
5 4 3 60
17. Introduction to Counting and Probability
Basic Counting Principles
Example 2
How many outcomes can you
have when you toss:
2 (head and tail)
a. One coin?
2x2=4
b. Two coins?
2x2x2=8
c. Three coins?
18. Introduction to Counting and Probability
Basic Counting Principles
Example 3
How many outcomes can you have
when you toss:
6
a. One die?
6 x 6 = 36
b. Two dice?
6 x 6 x 6 = 216
c. Three dice?
6 x 2 = 12
d. A die and a coin?
19. Fundamental Principle of Counting
Basic Counting Principles
Example 4
How many ways can you answer a
a. 20-item true or false quiz?
20 x 2 = 40
b.
20-item multiple choice test, with
choices A, B, C, D?
20 x 4 = 80
20. Fundamental Principle of Counting
Basic Counting Principles
Example 5
How many three-digit numbers
can form from the digits 1, 2, 3, 4
if the digits
a. can be repeated? 4 x 4 x 4 = 64
b. cannot be repeated?
4 x 3 x 2 = 24
21. Fundamental Principle of Counting
Basic Counting Principles
Example 7
How many three-digit EVEN
numbers can form from the digits
1, 2, 3, 4 if the digits
a. can be repeated? 4 x 4 x 2 = 32
b. cannot be repeated?
3 x 3 x 2 = 12
22. Introduction to Counting and Probability
Basic Counting Principles
Example 8
How many three-digit numbers
can form from the digits 0,1, 2, 3, 4
if the digits
a. can be repeated? 4 x 5 x 5 = 100
b. cannot be repeated?
4 x 4 x 3 = 48
23. Introduction to Counting and Probability
Check your understanding
1.
2.
3.
How many subdivision house numbers can
be issued using 1 letter and 3 digits?
How many ways can you choose one each
from 10 teachers, 7 staff, and 20 students
to go to an out-of-school meeting?
How many 4-digit numbers can be formed
from the digits 1 2, 3, 4, 5 if the digits cannot
be repeated?
Answers:
1. 26 x 10 x 10 x 10 = 26,000
2. 10 x 7 x 20 = 1,400
3. 5 x 4 x 3 x 2 = 120
24. Introduction to Counting and Probability
Basic Probability
Probability is a relative measure
of expectation or chance that
an event will occur.
Question: How likely is an event to
occur based on all the possible
outcomes?
25. Introduction to Counting and Probability
Basic Probability
Computing Probability
The probability p than an event can
occur is the ratio of the number of ways
that the event will occur over the
number of possible outcomes S.
number of ways that a certain event will occur
p
number of possible outcomes
26. Introduction to Counting and Probability
Basic Probability
Example 1
There are two outcomes in a toss of
a coin – head or tail. Thus, the
probability that a head will turn up in
a coin toss is 1 out of 2; that is,
number of heads
1
p
number of possible outcomes 2
27. Introduction to Counting and Probability
Basic Probability
Example 2
There are 6 outcomes in a roll of die.
What is the probability of getting a
a. 6?
One out of 6: p = 1/6
b. 2 or 3?
Two out of 6: p = 2/6 or 1/3
c. odd number? Three out of 6: p = 3/6 or 1/2
Zero out of 6: p = 0/6 or 0
d. 8?
Letter d is an IMPOSSIBLE EVENT
28. Introduction to Counting and Probability
Basic Probability
Example 3
A bowl has 5 blue balls, 6 red balls, and 4 green
balls. If you draw a ball at random, what is the
probability that you’ll
5 out of 15: p = 5/15 = 1/3
a.
get a blue ball?
6 out of 15: p = 6/15 = 2/5
b.
get a red ball?
4 out of 15: p = 4/15
c.
a green ball?
d.
not get a red ball? 5 + 4 = 9 out of 15: p = 9/15 or 3/5
e.
get a red or green ball?
6 + 4 = 10 out of 15:
p = 10/15 = 2/3
29. Introduction to Counting and Probability
Check your understanding
1.
2.
3.
Two coins are tossed. What is the
probability of getting two tails?
In a game of Bingo, what is the probability
that the first ball comes from the letter G?
All the three-digit numbers formed by using
the digits 1, 2, 3, and 4 without repeating
digits are put in a bowl. What is the
probability that when a number is drawn, it
is odd?
30. Introduction to Counting and Probability
Check your understanding
1. 1 out of 8:
p = 1/8
2. 10 out of 75: p = 10/75 = 2/15
3. Number of 3-digit numbers: 4 x
3x2
= 24
Number of odd 3-digit numbers:
3 x 2 x 2 = 12
p = 12/24 = 1/2