it is for easy understanding mathematics

1. Grade 10 Mathematics
Prepared by:
Kenth Richard M. Romulo

3. BIRPYIBALTO
INTERSECTION
ETRIONINCTES
PROBABILITY

4. NIOUN
EVENTS
EVNETS
UNION

5. IRNPMEEEXT
SAMPLE
SALPME
EXPERIMENT

6. OARDMN
OUTCOMES
OMOCTUSE
RANDOM

7. NEHCCAS
VENN DIAGRAM
ENNV IAMRADG
CHANCES

9. STATISTICS AND PROBABILITY

10. The chance that something will happen - how likely it is
that some event will happen.

11. SAMPLE SPACE
A sample space is the set of all possible
outcomes of an experiment
OUTCOME
Each element of a sample space is called an
outcome or a sample point.
EVENT
Any subset of a sample space is an event(E).

12. EXPERIMENT: In rolling a die.
A die has faces numbered 1 to 6.
Hence, S = {1,2,3,4,5,6}

13. EXPERIMENT: In rolling a die.
A die has faces numbered 1 to 6.
Hence, S = {1,2,3,4,5,6}
Event (E): getting an even number
E = {2, 4, 6}

14. SIMPLE EVENT
A simple event consists of a single outcome that cannot be
further broken down into smaller events. Tossing a single
coin will produce either head or tail [H,T].
COMPOUND EVENT
A compound event is any event combining two or more
simple events. The event that at least one head appears in
tossing a coin twice is a compound event.

15. P (E) =
𝑛 ( 𝐸 )
𝑛 ( 𝑆 )
Where,
E = event
S = Sample Space
n ( E ) = Number of Events
n ( S ) = Total Number or values in the Sample Space

16. Experiment: throwing a die
Event (E): getting an even number
Solution:
A die has faces numbered 1 to 6.
Hence, S = {1,2,3,4,5,6}
E = {2, 4, 6}
P (E) =
𝑛 ( 𝐸 )
𝑛 ( 𝑆 )
P (E) =
3
6
P (E) =
1
2

17. Experiment: Tossing a coin
Event (E): getting a head
Solution:
A coin has two faces.
S = {H, T} n(S) = 2
E = {H} n(E) = 1
P (E) =
𝑛 ( 𝐸 )
𝑛 ( 𝑆 )
P (E) =
1
2

18. AND PROBABILITY
And means that the
outcome has to
satisfy both
conditions at the
same time.
OR PROBABILITY
Or means that the
outcome has to satisfy
one condition, or the
other condition, or both
at the same time.

19. Find the probability of “getting a 6 and a 1”
when two dice are rolled is an event consisting
of (1, 6), (6, 1) as outcomes.

20. The first die falls in 6 different ways and the second die also falls
in 6 different ways.
Thus, using the fundamental counting principle, the
number of outcomes in the sample space is 6x6 or 36. The
outcomes in the sample space are: {(1, 1), (1, 2), (1, 3), (1,
4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3),…,(6, 5), (6, 6)}.
Take note that “getting a 6 and a 1” when two dice are
rolled is a compound event consisting of {(1, 6), (6, 1)} as
outcomes.

21. n (S) = 36
n (E) = (6,1), (1, 6)
P (E) =
𝑛 ( 𝐸 )
𝑛 ( 𝑆 )
P (E) =
2
36
P (E) =
1
18

23. Find the probability if two dice are tossed, what is
the probability that the sum will be 4 or 5?
Solution:
Let E1 be the event that the sum of 4 will come out
and E2 be the sum of 5 will come out. The number
of elements in the sample space S, n(S), is equal to
6x6=36 or n(S) = 36.

24. In this problem, we have the following outcomes for
each event:
E1 = {(2,2), (1,3), (3,1)}
E2 = {(2,3), (3,2),(1,4),(4,1)}
n(E) = 3+4 = 7
P (E) =
𝑛 ( 𝐸 )
𝑛 ( 𝑆 )
P (E) =
7
36

25. Let E1 and E2 be the two successive events in a sample space
S. If the outcome of E2 is affected by the prior occurrence of
E1, then E2 and E1 are dependent events. However, if the
outcome of E2 is not affected by the prior occurrence of E1,
then E1 and E2are independent events.
We use the symbol P (E2 / E1) as notation to read as, “the
probability of the occurrence of E2 on the condition that E1
has already occurred.

26. A man draws a card twice from an ordinary deck wherein
the first event (E1) is getting a red card and getting black
king is the second event (E2) with the following
conditions:
A. The first card was returned before the second card is
drawn.
B. The first card was not returned.

27. A. Since the first card was replaced then the events
are independent. This means the event of getting
a black king is not affected by the prior
occurrence of the event of drawing a red card.
B. Since there was no replacement after the first
event E1 has been performed, then second event
E2 is affected and is dependent to E1.

28. Since events are sets, can be combined to form new
events by using the set operation of union. It can be
illustrated by means of Venn diagram.

29. A die is tossed.
Sample Space (S) = {1, 2, 3, 4, 5, 6}
Let A = event that an odd number occurs
B = event that a number greater than 4 occurs
Determine the elements of A and B. then, find A
U B and draw a Venn diagram to illustrate it.
Solution:
A = {1, 3, 5}
B = {5, 6}
A U B = {1, 3, 5, 6}

30. Two events A and B intersect if there are
elements common to both A and B. It is
denoted by A ∩ B.

31. A fair die is rolled.
The sample is S = {1, 2, 3, 4, 5, 6}
Let A = event “even number turns up”
B = events “the number that turns up is greater
than
2”
Find A n B and draw a Venn diagram to illustrate it.

32. A = {2, 4, 6}
B = {3, 4, 5, 6}
Thus, A ∩ B = {4, 6}

33. The cardinality of a set refers to the number of
elements of the set. The cardinality of set A is
denoted by |A|.

34. Let A = {2, 4, 6, 8, 10}. Find |A|?
Solution:
Since there are 5 elements in set A, therefore,
|A| = 5.

35. The sample space S consist of the six possible
outcomes of die tossed once.
S = {2, 3, 4, 5, 6, 7}. Find |S|?
Solution:
Sample space S contains 6 outcomes, therefore,
|S| = 6.

36. Directions: Choose the letter of the correct answer.
1.What is a subset of a sample space called?
a) Element
b) Event
c) Set
d) Cardinality
2. How is union of sets A and B denoted?
a) AB
b)A U B
c)A
d)A+B

37. 3. The union and intersection can be illustrated by means of ____________.
a) Fish bone
b)Venn Diagram
c)Flow Diagram
d)Organizational Chart
4. What is described if there are elements common to both events A and B?
a)Complement
b)Union
c)Cardinality
d)Intersection

38. 5. What is the probability that an event can never be happened?
a) 0
b)between 0 and 1
c)1
d)cannot be determined
6. Which type of events is affected by prior occurrence of the other event?
a)Dependent
b)Mutually exclusive
c)Independent
d)Simple

39. 7. Which condition affects the drawing of cards twice from an ordinary
deck? The ___
a) first card was not returned
b)first card was returned
c)second card is returned
d)Second card is drawn
8. Which is NOT true about compound event?
a)The event that at least one head appears in tossing a coin twice.
b)Then probability of “getting a 6 and a 1” when two dice are rolled.
c)It is a combination of two simple events.
d)It is the intersection of two events.

40. 9. What is described when an event contains exactly one sample point or
outcome?
a) Simple event
b)Independent event
c)Compound event
d)Mutually exclusive
10. What term describes the manner of drawing cards, tossing, or rolling of a
coin and a die repeatedly?
a)Outcome
b)Sample point
c)Experiment
d)Event

41. 11. Which shade illustrates the union of three events A, B and C in a sample
space?

42. 12. What is the probability of getting 2 or 6 in rolling a die?
a) 5/6
b) 1/6
c) 2/3
d) 2/6 or 1/3
13. What is the intersection of events A and B if A are whole numbers up to 10
and B are numbers divisible by 3?
a) A ∩ B = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
b) A ∩ B = { 0, 2, 4, 6, 8, 10 }
c) A ∩ B = { 0, 1, 3, 5, 7, 9 }
d) A ∩ B = { 3, 6, 9 }

43. 14. Picking a king of hearts in a deck of cards
a) Simple event
b) Compound event
15. Which of the following Venn diagrams illustrates the intersection of events A
and C?