Introduction to probability concepts.pptx

BIRPYIBALTO
INTERSECTION
ETRIONINCTES
PROBABILITY

IRNPMEEEXT
SAMPLE
SALPME
EXPERIMENT

OARDMN
OUTCOMES
OMOCTUSE
RANDOM

NEHCCAS
VENN DIAGRAM
ENNV IAMRADG
CHANCES

LESSON 4.4 – A
ILLUSTRATING EVENTS
define an experiment,
outcome, sample space and
event;
 illustrate an experiment,
outcome, sample space and
event; and
 find the probability of the
given events.

EXPERIMENT
An experiment is an activity that
produces results.
tossing a coin
spinning a wheel
drawing a card
rolling a die

EXPERIMENT
- an activity that produces
results.
OUTCOME
- a result of an experiment.
SAMPLE SPACE
- collection of all outcomes
of the experiment.
EXAMPLES
1. Find the sample space
of tossing one coin.
S = { H, T }
- subset of a sample space.
EVENT
H
(head)
T
(tail)
Therefore, there are 2 possible
outcomes if we tossed one coin.
Event (A): getting a head
A = { H }
OUTCOMES

Therefore, there are 6 possible outcomes if we
rolled one die.
rolled one die.
2. Find the sample
space of rolling a
die.
S = { 1, 2, 3, 4, 5, 6 }
3. Find the sample space of
tossing a coin three times.
6 faces with
numbers 1 to 6
outcomes if we rolled one die.
S = { HHH, HHT, HTH, HTT,
THH,THT,TTH,TTT }
outcomes if we tossed one coin
three times.
1st 2nd 3rd
toss toss toss
1st 2nd 3rd
toss toss toss
Event (A): getting an even number
A = { 2, 4, 6 }
OUTCOME
S

Let B stands for “boy” and G stands for “girl”.There
will be an order of the
outcomes.
S = { BBB, BBG, BGB, BGG, GGG, GGB, GBG, GBB }
Since their first child is a girl, we will only get the
outcomes with G as the first child.
S = { GGG, GGB, GBG, GBB }
Therefore, there are 4 possible outcomes if it is
known that their first child is a girl.
4. A couple has three
children and if it is known
that their first child is a girl.
Find the sample space of
their possible children.
order of the outcomes for three
children is:
{ BBB, BBG, BGB, BGG, GGG, GGB,
GBG, GBB }
G
(girl)
B
(boy)
Since their first child is a girl, we
will only get the outcomes with
G as the first child.
outcomes if it is known that their
first child is a girl.

tossed a die and a coin.
5. A die and a coin are
tossed. Find the possible
outcomes of the experiment.
S = { 1H, 2H, 3H, 4H, 5H, 6H,
1T, 2T, 3T, 4T, 5T, 6T }
outcomes if we tossed a die
and a coin.

EXAMPLES:
- tossing a coin
- rolling a die
SIMPLE EVENT
TYPES OF
EVENTS
- can only happen in one
way
- has a single outcome
- combination of two or
more simple events
EXAMPLES:
- tossing a coin
- rolling a die
EXAMPLE:
The event that at least
one head appears in
tossing a coin twice
SIMPLE EVENT COMPOUND
EVENT

The
probability of event A, denoted by P(A), is the
probability that the outcome of the
experiment is contained in A.
- the possibility or chance that
an event will happen
- how likely it is that some
event will happen.
PROBABILITY The probability of event A,
denoted by P(A), is the
probability that the outcome of
the experiment is contained in A.
P(A) =
number of favorable
outcomes
total number of outcomes
P(A) = number ofevents
total number or values in the sample space
Note:
The highest result in probability is 1 while the
lowest is 0.
It can be expessed in decimal or fraction form.

EXAMPLES:
- tossing a coin
- rolling a die
Find the probability of getting a number less than 5
in rolling a die.
EXAMPLES
1. Find the probability
of getting a tail in
tossing one coin.
of getting a number
less than 5 in rolling a
die.
S = { H, T}
A = { T }
P(A) = ½ or 0.5
S = { 1, 2, 3, 4, 5, 6}
A = {1, 2, 3, 4}
P(A) = 4/6 = 2/3 or 0.67

EXAMPLES:
- tossing a coin
- rolling a die
in rolling a die.
of getting at least 2
heads in tossing a coin
three times.
S = { HHH, HHT, HTH, HTT,
THH,THT,TTH,TTT }
A- = { HHH, HHT, HTH,THH }
P(A) = 4/8 = 1/2
of getting a blue ball
P(blue) = 5/12

EXAMPLES:
- tossing a coin
- rolling a die
in rolling a die.
of getting a red card.
(52 cards)
of landing on a yellow.
P(red) = 26/52 = 1/4
P(yellow) = 3/8

EXAMPLES:
- tossing a coin
- rolling a die
in rolling a die.
A. Identify the term described i
each statement below.
1. It refers to an activity that
produces results.
2. It is a subset of a sample space.
3. It is the collection of all
outcomes of the experiment.
4. It is a combination of two or
more simple events.
5. It is a result of an experiment.
B. Solve the following problems.
1. A bag contains six identical
balls, two red balls, three blue
balls and one yellow ball. Find
the probability of getting a
yellow ball.
2. From a deck of an ordinary
playing cards, find the
probability of getting a
heart card.
3. A die is rolled once. Find the
probability of getting an even
number.
SW 4.10

Lesson 4. 4 - B
define independent and dependent events;
find the probability of independent and dependent events

Situation 1:
A ball is drawn at random and
the color is noted and then put
back inside the box.
Then another ball is drawn at
random.
Find the probability that the
first ball is green and the
second is yellow.
Situation 2:
Suppose that two balls are
drawn one after the other
without putting back
the first ball. Find the
probability that the first ball
the first ball is green and
the second is yellow
Which one has a better chance of winning?
4 green balls
5 blue balls
3 yellow balls

Situation 1: Consider a bag that
contains 4 green balls, 5 blue balls
and 3 yellow balls. A ball is drawn
at random and the color is noted
and then put back inside the box.
Then another ball is drawn at
random. Find the probability that
the first ball is green and the
second is yellow
Situation 1: Consider a bag that
contains 4 green balls, 5 blue balls and
3 yellow
balls. Suppose that two balls are drawn
one after the other without putting back
the first ball. Find the probability that
the first ball the first
ball is green and the second is yellow
the second draw is NOT
AFFECTED by the probability of
the first draw, since the first ball is
put back inside the box
(REPLACED)
the ball was not put back in the
box (NOT REPLACED),thus
the drawing of two balls would
be dependent
INDEPENDENT EVENTS DEPENDENT EVENTS

Two events are independent if
the occurrence of one of the
events gives us no information
about whether the other event
will occur,and the events have no
influence on each other.
If two events,A and B,are
independent,then the
probability of both events
occurring is the product of the
probability of A and the
probability of B.
When the outcome of one event
affects the outcome of the other
event,they are said to be
dependent events.
If two events,A and B,are
dependent,then the probability
of both events occurring is the
product of the probability of A
and the probability of B after A
occurs.
INDEPENDENT EVENTS DEPENDENT EVENTS
𝑃( ) = ( )
𝐴 𝑎𝑛𝑑 𝐵 𝑃 𝐴 ∙ (
𝑃 𝐵 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔
)
𝐴
𝑃( ) = ( )
𝐴 𝑎𝑛𝑑 𝐵 𝑃 𝐴 ∙ ( )
𝑃 𝐵

1. A box that contains 10 red balls, 6 blue balls and 4 yellow
balls. A ball is drawn at random and the color is noted and
then put back inside the box.Then another ball is drawn at
random. Find the probability that the first ball is red and the
second is yellow.
EXAMPLES
2. A box that contains 10 red balls, 6 blue balls and 4 yellow
balls. Suppose that
two balls are drawn one after the other without putting back
the first ball. Find the
probability that the first ball is red and the second is yellow.

3. A bag of candies contains 9 strawberry, 6 coffee, 7
orange, and 8 caramel Candies.What is the
probability of randomly choosing a strawberry
candy, eats it, Randomly choosing an orange candy,
eats it, and then randomly choosing a caramel
candy?
4. A bag of candies contains 9 strawberry, 6 coffee, 7
orange, and 8 caramel candies.What is the
probability of randomly choosing a strawberry
candy, replacing it, randomly choosing an orange
candy, replacing it, and then randomly choosing a
caramel candy?

5. A basket contains 2 apples, 4 bananas, 3 oranges and 6
mangoes. Ana randomly chooses one fruit, replaced it, and
chooses another fruit.What is the
probability that he chose an orange and then another
orange?
mangoes. Ana andomly chooses one fruit, eats it, and
chooses another fruit.What is the probability
that he chose an orange and then another orange?

1. A box that contains 10 red balls, 6 blue balls and 4 yellow balls. A ball is
drawn at random and the color is noted and then put back inside the box.
Then another ball is drawn at random. Find the probability that the first ball is
red and the second is yellow.

2. A box that contains 10 red balls, 6 blue balls and 4 yellow balls. Suppose that
two balls are drawn one after the other without putting back the first ball. Find
the probability that the first ball is red and the second is yellow.

3. A bag of candies contains 9 strawberry, 6 coffee, 7 orange, and 8 caramel
Candies.What is the probability of randomly choosing a strawberry candy,
eats it, Randomly choosing an orange candy, eats it, and then randomly
choosing a caramel candy?

4. A bag of candies contains 9 strawberry, 6 coffee, 7 orange, and 8 caramel
candies.What is the probability of randomly choosing a strawberry candy,
replacing it, randomly choosing an orange candy, replacing it, and then
randomly choosing a remedy candy?

mangoes. Ana randomly chooses one fruit, replaced it, and
chooses another fruit.What is the
probability that he chose an orange and then another orange?

6. A basket contains 2 apples, 4 bananas, 3 oranges and 6 mangoes.
Ana andomly chooses one fruit, eats it, and chooses another fruit.
What is the probability
that he chose an orange and then another orange?

SW 4.___
Instructions: Determine whether the events are independent or dependent.
Then, find the probability.
1. Maria has 5 black pens, 2 blue pens and 3 red pens in her bag. She randomly
picks two pens out of her bag.What is the probability that Maria chose two
black pens, if she did not put back the first pen before choosing another pen?
2. A bag contains 8 blue marbles, 2 green marbles, 6 pink marbles, and 4 red
marbles. A marble is randomly selected, returned, and a second marble is
randomly selected. Find the probability of selecting a pink marble, then a
green marble.
3. A rental car agency has 10 red cars, 11 white cars, and 9 gray cars. John
rents a car, returns it because it has damaged, and get another car.What is
the probability that John is given a red car and then a gray car?

PROBABILITY OF UNION AND INTESECTION
OF EVENTS
Lesson 4. 4 - C
 illustrates union and intersection of events; and
 perform the operations union, intersection and complement of events
 Illustrate probability of two events using Venn Diagram

VENN DIAGRAM
Venn diagram is a diagram
that uses circles to
represent sets, in which the
relations between the sets
are indicated by the
arrangement of the circles.

Intersection of Events Union of Events Complement of an Event
A
A’
The intersection of events A and B
-the set of all sample points in
the sample space that are in A
and B.
- denoted as 𝐴 ∩ 𝐵.
The union of events A and B
- the set of all sample points in
the sample space that are in A
or in B or both.
- denoted as .
𝐴 ∪ 𝐵
The complement of an event
-the set of all outcomes that are
NOT in the event.
𝐴 ∩ 𝐵 is the event
that both events
A and B occur.
𝐴 ∪ 𝐵 is the event that
either event A or event B
occurs, or both events occur.
Example:
the complement of an event A
is the set of all outcomes
which are not in A.

EXAMPLE 1
The beverage that the people prefer to drink in the morning are
shown in the Venn diagram below.
a. How many people drink tea in the morning?
b. How many people drink coffee in the morning?
c. How many people drink only tea in the morning?
d. How many people drink only coffee in the morning?
e. How many people drink both coffee and tea?
f. How many people drink coffee or tea?
g. How many people do not drink coffee or tea?
h. How many people took the survey?

EXAMPLE 2 A.Write the elements of the following:
1. U =
2. A =
3. B =
4. A ∪ B =
5. A B =
∩
6. A’ =
B. Find the probability of the following:
7. P(A)
8. P(B)
9. P(A ∪ B )
10. P(A B)
∩
C. Illutrate P(A ∪ B ) using Venn Diagram
7 9
Illustration of A ∪ B

EXAMPLE 3
The Venn diagram below shows the probabilities of
grade 10 students joining either soccer (S) or
basketball (B).
Use the Venn diagram to
find the probabilities.
a. P(B)
b. P(S)
c. P(B S)
∩
d. P(B S)
∪

SW 4.___
A.Write the elements of the following:
1.A =
2.B =
3.A ∪ B =
4.A B =
∩
B. Find the probability of the following:
7. P(A)
8. P(B)
9. P(A ∪ B )
10. P(A B)
∩
C. Illutrate P(A ∪ B ) using Venn Diagram

Lesson 4. 4 - D
Finding the Probability of (𝑨
)
∪ 𝑩
- differentiate mutually exclusive events from non-mutually
exclusive events
- find the probability of the union of two events.

MUTUALLY EXCLUSIVE
EVENTS
(DISJOINT EVENTS)
NON-MUTUALLY EXCLUSIVE
EVENTS
(INCLUSIVE EVENTS)
𝑷( ) = ( ) +
𝑨 ∪ 𝑩 𝑷 𝑨
( )
𝑷 𝑩
𝑷( ) = ( ) + ( ) (
𝑨 ∪ 𝑩 𝑷 𝑨 𝑷 𝑩 − 𝑷 𝑨
)
∩ 𝑩
-events that cannot occur at
the same time
- does not have
intersection
-events that can occur at
the same time;
- has intersection

FIND THE PROBABILITY OF THE FOLLOWING:
1. a King ( ) = _____
𝑃 𝐾
2. a Heart ( ) = _____
𝑃 𝐻
3. a Queen ( ) = _____
𝑃 𝑄
5. a Queen and Heart ( ) = _____
𝑃 𝑄 ∩ 𝐻
6. a King or Queen ( ) = _____
𝑃 𝐾 ∪ 𝑄
7. a Queen or Heart ( ) = _____
𝑃 𝑄 ∪ 𝐻

8. In a class of 40 students, 13 are enrolled in Chinese for their elective foreign
language class and 15 are enrolled in Spanish. If each student can only enroll in
exactly one foreign language, what is the probability that a student from the said
class is enrolled in Chinese ( ) or Spanish ( )?
𝐶 𝑆
9. In a class of 40 students, 13 are enrolled in Chinese for their elective foreign
language class, 15 are enrolled in Spanish, and 8 are enrolled in both Chinese and
Spanish.What are the chances that a student from the said class is enrolled in
Chinese ( ) or Spanish ( )?
𝐶 𝑆
10. When rolling a die, what is the probability of rolling a number less than 3
( )
𝐿 or a number greater than 3 ( )?
𝐺
11. When rolling a die, what are the chances of rolling a number less than five
( )
𝐿 or an even number ( )?
𝐸

A. Instructions: Identify whether or not these events are
mutually exclusive (ME) or not mutually exclusive (NME).
1.The event of getting a perfect score in a test and the event
of getting a
passing score
2.The event that a baby is born with blood type A and the
event that the
same baby is born with blood type O
3.The event of drawing a club from a deck of cards and the
event of drawing
a diamond from a deck

4. a Queen and Jack (
𝑃 𝑄 ∩ J) = _____
5. a Jack or Queen (
𝑃 J ) = _____
∪ 𝑄
6. a Queen or Diamond (
𝑃 𝑄 ∪ D) = _____
B. FIND THE PROBABILITY OF THE FOLLOWING:
7. Of the 50 presidential candidates for election, 32 are
females ( ) and 9 are Cebuanos ( ). All Cebuano
𝐹 𝐶
candidates are males.What is the probability that a
female or a Cebuano is elected as president?
8. Of the 50 presidential candidates for election, 32 are
females ( ) and 9 are Cebuanos ( ).Three of the
𝐹 𝐶
Cebuano candidates are males.What is the probability
that a female or a Cebuano is elected as president?

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