Transcript Chapter 5: probability

Mr. Mark Anthony Garcia, M.S.
Mathematics Department
De La Salle University
Probability
Probability is the likelihood that a certain
event will happen. The probability value
may be written in relative frequency form
(A value between 0 and 1) or in
percentage form (relative frequency times
100) or in fraction form (numerator is less
than or equal to denominator)
Null Event
A null event or null space is an event that
is impossible to happen. The null event is
denoted by the symbol Ø.
Example 1: Null Event
The event of obtaining the number 7
when a die is thrown.
 The event of obtaining a heart card
which is black in a deck of playing cards.
 The event that there will be no January
1 in a year.

Probability: Properties
The probability of an event A, denoted
by, P(A), is always between 0 and 1,
inclusive.
 The probability of the null event ø,
denoted by, P(ø) is 0.
 The probability of the sample space S,
denoted by, P(S) is 1.

Priori Probability
Let E be an event which is a subset of the
sample space S. The priori probability is
the theoretical probability that event E will
𝑛
happen. It is given by the formula 𝑃 𝐸 =
𝑁
where n is the number of sample points in
the event E and N is the number of sample
points in the sample space S.
Example 2: Priori Probability
Consider the experiment of tossing a
coin.
 The sample space is S = {H, T}.
 Let E be the event of obtaining a head.
Then E = {H}.
 The priori probability of the event E is
𝑛
1
𝑃 𝐸 = = = 0.5 or 50%.

𝑁
2
Posteriori Probability
Consider
a
particular
experiment
containing N trials. Let E be an event
which is a subset of the sample space S of
the experiment. The posteriori probability
𝑛
of the event E is given by 𝑃 𝐸 = where
𝑁
n is the number of times that sample
points of E occurred in N trials.
Example 3: Posteriori Probability
Consider the experiment of tossing a
coin.
 Let N = 20 be the number of tosses.
 Let E be the event of obtaining a head.
Then E = {H}.
 Consider the results table in the next
slide.

Example 3: Posteriori Probability
1. H
2. H
3. T
4. T
5. H
6. T
7. H
8. H
9. T
10. H
11. T
12. T
13. H
14. T
15. H
16. H
17. T
18. H
19. T
20. H
Example 3: Posteriori Probability
No. of times H
occurred in first
N trials
First N trials
Posteriori
Probability
2
5
6
9
11
4
8
12
16
20
0.5
0.625
0.5
0.5625
0.55
Probability Formula 1: Basic Rule
Let A be an event. The probability of A is
given by
𝑛
𝑃 𝐴 =
𝑁
where n is the number of elements of
event A and N is the number of elements
of the sample space S.
Example 4: Probability Formula 1
A coin is tossed twice. What is the
probability that at least 1 head occurs?
 The sample space is S = {HH, HT, TH,
TT}.
 The event A is the event that at least
one head occurs in the two tosses of the
coin.

Example 5: Probability Formula 1
A die is rolled once, what is the
probability that an even number occurs?
 The sample space is S = {1,2,3,4,5,6}.
 The event A is the event that an even
number occurs in the roll.

Example 6: Probability Formula 1
If a card is drawn from an ordinary deck,
find the probability that a heart card is
drawn.
 The sample space is the set of 52
playing cards.
 The event A is the event that the card
drawn is a heart card.

Example 7: Probability Formula 1
A mixture of candies contains 6 mints, 4
toffees and 3 chocolates. If a person
makes a random selection of one of
these candies, find the probability of
getting a mint.
 The event A is the event of getting a
mint.


Hence, 𝑃 𝐴 =
46.15%.
𝑛
𝑁
=
6
13
= 0.4615
or
Set Operations
The union of two events A and B, denoted
by, 𝐴 ∪ 𝐵, is the event containing elements
from A or from B.
 The intersection of two events A and B
denoted by, denoted by 𝐴 ∩ 𝐵, is the event
containing elements from both A and B.
 The complement of an event A, denoted by
𝐴′ or 𝐴𝑐 , is the event containing elements
from the sample space S that are not in A.

Example: Set Operations
Let S = {1,2,3,4,5,6} be the sample
space.
 Let A = {1,4,5} and B = {4,6}.
 Then 𝐴 ∪ 𝐵 = {1,4,5,6}, 𝐴 ∩ 𝐵 = {4} and
𝐵𝐶 = {1,2,3,5}.

Mutually Exclusive Events
Two events A and B are mutually exclusive
if 𝐴 ∩ 𝐵 = ∅. This means that the event
𝐴 ∩ 𝐵 is impossible to happen.
Example 8: Mutually Exclusive
Events
Let A be the event that a heart card is
drawn and B be the event that a black
card is drawn. Then 𝐴 ∩ 𝐵 = ∅. Thus, A
and B are mutually exclusive events.
 Let A be the event that team X loses
game 3 of a best-of-three championship
series. Let B be the event that team X
wins the championship. Hence, A and B
are mutually exclusive events.

Probability Formula 2: Additive
Rule
Let A and B be two events. Then
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵).
 If the events A and B are mutually
exclusive, then 𝑃 𝐴 ∩ 𝐵 = 𝑃 ∅ = 0 .
Thus, 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 .

Example 9: Probability Formula 2
The probability that a student passes
Algebra is 2/3, and the probability that he
passes English is 4/9. If the probability of
passing at least 1 subject is 4/5, what is
the probability that he will pass both
subjects?
Example 10: Probability Formula 2
What is the probability of getting a total of
7 or 11 when a pair of dice is tossed?
Probability Formula 3:
Complementary Rule
Let A be an event. Then 𝑃 𝐴 + 𝑃(𝐴𝑐 )=1.
 The probability of an event A plus the
probability of its complement is equal to
one.

Example 11: Probability with
Combinations
If 3 books are picked at random from a
shelf containing 5 novels, 3 books of
poems, and a dictionary, what is the
probability that
A.
B.
the dictionary is selected?
2 novels and 1 book of poems are
selected?
Example 12: Probability with
Permutations
If a permutation of the word “white” is
selected at random, find the probability
that the permutation
A.
B.
C.
Begins with consonant;
Ends with a vowel;
Has the consonants
alternating.
and
vowels
Example 13: Probability in Twoway tables
A random sample of 200 adults are
classified below according to sex and level
of education attained.
Elementary
Secondary
College
Total
Male
38
28
22
88
Female
45
50
17
112
Total
83
78
39
200
Example 13: Probability in Twoway tables
If an adult is selected at random, what is
the probability that
A. the adult is a male or has reached
elementary level only
B. the adult is a female a and reached up
to secondary level
C. the adult did not reach college level
Probability Formula 4:
Conditional Probability
The probability of an event B occurring
when it is known that some event A has
already occurred is called a conditional
probability, and is denoted by, P(B|A).
The symbol P (B|A) is read as “the
probability of B given A”. The probability of
B given A is defined by the equation
𝑃(𝐵 ∩ 𝐴)
𝑃 𝐵𝐴 =
𝑃(𝐴)
Example 14: Probability Formula 4
Suppose our sample space S is the set of
4th year high school student in a small
town who took the UPCAT college
entrance examination. Let us categorize
them according to sex and whether they
passed or not.
Example: Probability Formula 4
Passed
Male
30
Female
25
Total
55
Failed
110
65
175
Total
140
90
230
Example 14: Probability Formula 4
Find the probability that a student is a
male given that he passed the exam.
 Let M be the event that the student is a
male and let G be the event that the
student passed the exam.
A.

Then 𝑃 𝑀 𝐺 =
30
.
55
Example 14: Probability Formula 4
Find the probability that a student failed
the exam given that she is a female.
 Let F be the event that the student is a
female and H be the event that the
student failed the exam.
B.

Then 𝑃 𝐻 𝐹 =
65
.
90
Probability Formula 5:
Multiplicative Rule
If the events A and B can both occur, then
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃(𝐵|𝐴).
Example 15: Probability Formula 5
Suppose that we have a fuse box
containing 20 fuses, of which 5 are
defective. If 2 fuses are selected at
random and removed from the box in
succession without replacing the first, what
is the probability that both fuses are
defective?
Independent Events
Two events A and B are independent if the
occurrence of event A does not affect the
occurrence of event B.
Example 16: Independent Events
Consider the experiment of tossing a coin
twice. The first toss of the coin (event A)
does not affect the outcomes of the
second toss (event B). Thus, A and B are
independent events.
Example 16: Independent Events
Consider the departure (event A) and
arrival (event B) of airplanes. If the plane
departs on time, then it will probably arrive
on time. Therefore, A and B are dependent
events.
Probability Formula 6:
Independent Events
Two events A and B are independent if and
only if 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃 𝐵 .
Example 17: Probability Formula 6
A small town has one fire engine and one
ambulance available for emergencies. The
probability that the fire engine is available
when needed is 0.98, and the probability
that the ambulance is available when
called is 0.92. Find the probability that both
the ambulance and the fire engine will be
available.