Probability

Lessons in Probability for Mathematics 10

1. MATHEMATICS 10
Weeks 6-8 of Quarter 3

2. Lessons
✓Terms Related to Probability
✓Probability of a Simple or a Compound Event
??????(�)
✓Probability of a Union of Two Events
??????(�∪�)
✓Probability of Independent Events
??????(�∩�)
✓Conditional Probability
??????��

3. Some Phrases to Remember
▪at least a number (e.g.at least 3):
3, 4, 5, 6, 7, 8, 9, 10, 11, 12…
▪at most a number(e.g.at most 3):
0, 1, 2, 3 only
▪divisible by a number or a multiple of a number
(e.g.divisible by 4 or multiple of 4)
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
▪an odd number:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …
▪an even number:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
▪a prime number:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
▪acomposite number:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …
▪a non-prime and non-composite number:1 only

5. DEFINITION OF TERMS
•Probabilityrefers to the chance or
likelihood that an event will happen.
As a number, it lies between0(the event will
not happen) and 1(the event will happen).
0 or 0%
0.25 or
1
4
or 25%
0.75 or
3
4
or 75%
1 or 100%

6. DEFINITION OF TERMS
•Experimentis anything that is repeatedly
do where results may vary even conditions
are similar.
tossing a coin, tossing three coins,rolling a
die, rolling two dice, tossing a coin and a die,
drawing a card from a deck of cards

7. DEFINITION OF TERMS
•Sample Spaceis the set of all possible
outcomes in an experiment.
tossing a coin:
??????={H,T)
tossing two coins:
??????={HH, TT, HT, TH}
tossing three coins:
??????={HHH, TTT, HTH, THT, HHT, TTH, HTT, THH}
rolling a die: ??????={1,2,3,4,5,6}

8. rolling two dice:
??????={(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1),
(2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3),
(3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5),
(4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1),
(6,2), (6,3), (6,4), (6,5), (6,6)}
a family having 3 children:
??????={BBB, GGG, BGB, GBG, BBG, GGB, BGG, GBB}
tossing a coin and a die:
??????={H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

9. drawing from a standard deck of cards:

10. A standard deck of 52 cards comprises 13 ranks
in each of the four French suits: clubs( ), spades
( ), hearts( ), and diamonds( ).
Each suit includes an ace, number cards(2, 3, 4, 5,
6, 7, 8, 9, 10), and face cards(king, queen, jack).

11. DEFINITION OF TERMS
•Eventis any subset of a sample space.
getting a tail in tossing a coin:
??????={T}
getting one head and one tail in tossing twocoins:
??????={HT,TH}
getting at leasttwo heads when tossing three coins:
??????={HHH,HTH,HHT,THH}
having at most2 girls in a family with 3 children:
??????={BBB,BGB,GBG,BBG,GGB,BGG,GBB}
:

12. getting both even numbers when rolling two dice:
??????={(2,2), (2,4), (2,6), (4,2), (4,4), (4,6),(6,2),
(6,4), (6,6)}
getting a head and an even number when tossing a
coin and a die:
??????={H2, H4, H6}
getting a red face card from a deck of cards:
??????={ }

13. Experiment: selecting a letter from the word
SUPERCALIFRAGILISTICEXPIALIDOCIOUS
??????={S, U, P, E, R, C, A, L, I, F, R, A, G, I, L, I, S, T, I, C, E,
X, P, I, A, L, I, D, O, C, I, O, U, S}
??????=selectingaconsonant
??????={S, P, R, C, L, F, R, G, L, S, T, C, X, P, L, D, C, S}
More Examples: Sample Space (S) and Events (E)

14. Experiment: selecting a letter from the word
SUPERCALIFRAGILISTICEXPIALIDOCIOUS
??????={S, U, P, E, R, C, A, L, I, F, R, A, G, I, L, I, S, T, I, C, E,
X, P, I, A, L, I, D, O, C, I, O, U, S}
??????=selectingaconsonant
??????={S, P, R, C, L, F, R, G, L, S, T, C, X, P, L, D, C, S}
More Examples: Sample Space (S) and Events (E)

15. Experiment: picking a ball from a box containing
20 balls numbered 1 to 20
??????={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20}
??????=pickingacompositenumber
??????={4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20}
More Examples: Sample Space (S) and Events (E)

16. Experiment: tossing two coins
??????={HH, TT, HT, TH}
??????=gettingaheadandatail
??????={HT, TH}
Experiment: tossing three coins
??????={HHH, TTT, HTH, THT, HHT, TTH, HTT, THH}
??????=gettingatleast2heads
??????={HHH, HTH, HHT, THH}
Experiment: rolling a die
??????={1,2,3,4,5,6}
??????=gettinganoddnumber
??????={1,3,5,}
More Examples: Sample Space (S) and Events (E)

17. Experiment: rolling two dice
??????={(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2),
(2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4),
(3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2),
(6,3), (6,4), (6,5), (6,6)}
E = the first die shows an even number, and the
second die shows a number divisible by 3
??????=(2,3), (2,6), (4,3), (4,6), (6,3), (6,6)}
More Examples: Sample Space (S) and Events (E)

18. Experiment: choosing a child from a family
having 3 children
??????={BBB, GGG, BGB, GBG, BBG, GGB, BGG, GBB}
??????=gettingatmost2girls
??????={BBB, BGB, GBG, BBG, GGB, BGG, GBB}
Experiment: tossing a coin and a die
??????={H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
??????=gettingatailandaprimenumber
??????={T2, T3, T5,}
More Examples: Sample Space (S) and Events (E)

19. Experiment:
drawing from a standard deck of cards
??????=drawingablacknumbercard
??????=drawingan ace
??????=drawingaclub
??????=drawingaredfacecard
??????=drawingaredcard
??????=drawingajack
??????=drawinga3
??????=drawingaspade
??????=drawingablackking
??????=drawingaredqueen
??????=drawinganumbercard
(52)
(18)
(4)
(13)
(6)
(26)
(4)
(4)
(13)
(2)
(2)
(36)
More Examples: Sample Space (S) and Events (E)
Cardinality:

20. DEFINITION OF TERMS
•Sureeventis an event whose outcome must
occur. Probability is 1.
✓getting a counting number less than 7 when a die
is rolled
✓selecting a vowel letter from the word EUOUAE
•Impossible eventis an event whose
outcome must not occur. Probability is 0.
✓getting a 7 when a die is rolled
✓selecting a vowel letter from the word RHYTHMS

21. DEFINITION OF TERMS
•Sample pointis an outcome of an
experiment.
H, T, HH, TT, HT, TH, HHH, TTT, HTH, THT, HHT,
TTH, HTT, THH, 1, 2, 3, 4, 5, 6, (1,1), (1,2), (1,3),
(1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5),
(2,6), (3,3), (4,4), (5,5) (6,6), H1, H2, H3, H4, H5,
H6, T1, T2, T3, T4, T5, T6, BBB, GGG, BGB, GBG,
BBG, GGB, BGG, GBB
•Cardinalityis the number of outcomes in a
sample space or in an event.

22. Cardinality of a Sample Spaceor an Event:
•tossing a coin: 2
•tossing two coins: 4
•tossing three coins: 8
•rolling a die: 6
•rolling two dice: 36
•tossing a coin and a die: 12
•drawing a card from a deck of cards: 52
•getting one head in tossing twocoins: 2
•getting at leasttwo heads (tossing three coins): 4
•getting both even numbers (rolling two dice): 9
•having at most2 girls (family with 3 children): 7
•drawing a red face card from a deck of cards: 6

24. •Simple eventis an event with only one
outcome.
✓selecting a vowel letter in the word ANGRY
✓getting a non-prime and non-composite number
when rolling a die
•Compound eventis an event with more than
one outcome.
✓selecting a consonant in the word BIRD
✓getting two numbers with a sum greater than 5
when rolling two dice

25. PROBABILITY OF
simple and compound EVENTS
????????????=
numberoffavorableoutcomes
numberofpossibleoutcomes

26. 1.What is the probability of selecting a month of
the year with a letter “J” in its name?
????????????=
3
12
=
�
�
2.What is choosing a vowel letter in the word
COPYRIGHTABLE?
????????????=
�
��
PROBABILITY OF
simple and compound EVENTS

27. 3.A box contains 6 red marbles, 4 orange
marbles, 3 yellow marbles, 5 green marbles,
and 2 blue marbles. What is the probability of
drawing a greenmarble?
5
20
=
�
�
4.There are 45 students in a class. 20 of them
are boys. If a student is selected at random for
a field trip, what is the probability of selecting
a girl?
25
45
=
�
�

28. 5.There are 3 green chips, 5 orange chips, 2 red
chips, 4 yellow chips, and 6 blue chips in a jar.
A chip is to be drawn from the jar. What is the
probability that it is a color that is not
orange?
15
20
=
�
�
6.getting at least one tailwhen tossing two
coins?
�
�

29. 7.getting a perfect square numberwhen a die is
rolled?
2
6
=
�
�
8.having a girl as the youngest childin a family
with three children?
4
8
=
�
�

30. 9.getting two numbers with a sum greater than
8when rolling two dice?
10
36
=
�
��
10.drawing a black number cardfrom a deck?
18
52
=
�
��

32. UNION AND INTERSECTION
OF EVENTS
•Eventis a subset of the sample space.
•The union of eventsis the set of all outcomes
which belong to either first event or second
event or both. The union of events �and �is
denoted by �∪�(�or �).
•The intersection of eventsis the set of all
outcomes which belong toboth events. The
intersection of events �and �is denoted by
�∩�(�and �).

33. UNION AND INTERSECTION
OF EVENTS
A ball is picked from a box containing 20 balls
numbered from 1 to 20. Give the outcomes of
the two events.
�= the event of getting an even number
{2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
�= the event of getting a composite number
{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20}

34. UNION AND INTERSECTION
OF EVENTS
List the following:
1.outcomes found in either event �or set �or
both. (�∪�)
{2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20}
2.outcomes found both in event �and event �.
(�∩�)
{4, 6, 8, 10, 12, 14, 16, 18, 20}

35. Mutually exclusive events are events that do
not occur at the same time.
They are also called disjoint eventssince they
do not happen simultaneously.
MUTUALLY EXCLUSIVE EVENTS

36. Mutually Exclusive Events
Non Mutually Exclusive
Events
1.getting an odd number
or an even number when
rolling a die
2.picking a ball that shows a
number greater than 16
or a perfect square
number from a box of 20
balls numbered 1 to 20.
3.getting an aceor a
number cardwhen
drawing a card from a
standard deck of cards
1.getting an odd number or
a prime numberwhen
rolling a die
2.picking a ball that shows a
number less than 16or a
number divisible by 5
from a box of 20 balls
numbered 1 to 20.
3.getting an aceor a black
card when drawing a card
from a standard deck of
cards

37. A chip is picked from a bowl containing 25 chips
numbered from 1 to 25. Give the outcomes of the
two events.
�= the event of getting a number divisible by 5
{5, 10, 15, 20, 25}
�= the event of getting a number divisible by 6
{6, 12, 18, 24}
MUTUALLY EXCLUSIVE OR NOT?
They are mutually exclusive. There is no �∩�.

38. A ball is picked from a box containing 20 balls
numbered from 1 to 20. Give the outcomes of
the two events.
�= the event of getting an odd number
{1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
�= the event of getting a prime number
{2, 3, 5, 7, 11, 13, 17, 19}
MUTUALLY EXCLUSIVE OR NOT?
They are not mutually exclusive. It is because
there is �∩�={3,5,7,11,13,17,19}.

39. A card is drawn from a deck of 52 cards.
�= the event of drawing a spade
�= the event of drawing a face card
MUTUALLY EXCLUSIVE OR NOT?
They are not mutually exclusive. There is �∩�.

40. A card is drawn from a
standard deck of cards.
�= the event of
drawing an ace
�= the event of
drawing a number card
MUTUALLY EXCLUSIVE OR NOT?
They are mutually exclusive. There is no �∩�.

41. Two dice are rolled.
�= the event of getting two numbers whose sum
is greater than 6
{(1,6), (2,5), (2,6), (3,4), (3,5), (3,6), (4,3),
(4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5),
(5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
�= the event that the first numbers is odd, and
the second number is even
{(1,2), (1,4), (1,6), (3,2), (3,4), (3,6), (5,2),
(5,4), (5,6)
MUTUALLY EXCLUSIVE OR NOT?
They are not mutually exclusive.

42. PROBABILITY OF
A UNION OF TWO EVENTS
•If �and �are events in the same sample space,
then the probability of �or�occurring is:
??????��??????�=??????�∪�
=??????�+??????�−??????(�∩�)
•If �∩�is an empty set, then �and �are
mutually exclusive events:
??????��??????�=??????�∪�
=??????�+??????�

43. PROBABILITY OF
A UNION OF TWO EVENTS

44. PROBABILITY OF
A Union of two events
1.What is the probability of getting a number less
than 5ora number divisible by 3when a die is
rolled?
??????(�)getting a number less than 5 {1,2, 3, 4}
4
6
??????(�)getting a number divisible by 3 {3, 6}
2
6
??????�∩�getting both
less than 5 and divisible by 3 {3}
1
6
??????�∪�=
4
6
+
2
6
−
1
6
=
�
�

45. PROBABILITY OF
A Union of two events
2.What is the probability of getting a prime
numberoran odd numberin the rolling of a
die?
??????(�) getting a prime number {2, 3, 5}
3
6
??????(�) getting an odd number {1, 3, 5}
3
6
??????�∩�getting both prime and odd {3, 5}
2
6
??????�∪�=
3
6
+
3
6
−
2
6
=
4
6
=
�
�

46. PROBABILITY OF
A Union of two events
3.There are 20 balls inside a box. These balls are
numbered from 1 to 20. If a ball is drawn from
the box, what is the probability that it shows
an even numberora composite number?
??????(�) showing an even number
10
20
??????(�) showing a composite number
11
20
??????�∩�showing both even and composite
9
20
??????�∪�=
10
20
+
11
20
−
9
20
=
12
20
=
�
�

47. PROBABILITY OF
A Union of two events
4.In a deck of 52 cards, what is the probability of
drawing a red cardor a face card?
??????(�) drawing a red card
26
52
??????(�) drawing a face card
12
52
�∩�drawing both a red card
and a face card
6
52
??????�∪�=
26
52
+
12
52
−
6
52
=
32
52
=
�
��

48. PROBABILITY OF
A Union of two events
5.In a standard deck of cards, what is the
probability of drawing an aceor a black jack?
??????(�) drawing an ace
4
52
??????(�) drawing a black jack
2
52
??????�∩�drawing both an ace & a black jack0
??????�∪�=
4
52
+
2
52
=
6
52
=
�
��

50. PROBABILITY OF
Independent events
•If�and �are independent events, the
probability that both events �and�occur is the
product of their individual probabilities.
??????�??????????????????�=??????�∩�=??????�∙??????�
•Two events are independentif the occurrence or
non-occurrence of one event does not affectthe
probability of occurrence of the other event.

51. PROBABILITY OF
Independent events
otossing of a coin and a die
otossing of two coins
otossing of three coins
orolling of two dice
odrawing two balls, one at a time and with
replacementfrom a box
odrawing two cards, one at a time with
replacement, from a standard deck of cards

52. PROBABILITY OF
Independent events
1.Two letters will be chosen from the words
“word” and “games”. The first letter will come
from “word” and the second letter will come
from “games”. What is the probability that the
first letter is a consonantand the second letter
is a vowel?
??????(�)choosing a consonant letter (1
st
word: word)
�
�
??????(�)choosing a vowel letter (2
nd
word: games)
�
�
??????�∩�=
3
4
∙
2
5
=
6
20
=
�
��

53. PROBABILITY OF
Independent events
2.If two dice are rolled, what is the probability
that a number divisible by 3appears on the
first die and an even numberappears on the
second die?
??????(�)getting a number divisible by 3 (1
st
die)
2
6
or
�
�
??????(�)getting an even number (2
nd
die)
3
6
or
�
�
??????�∩�=
1
3
∙
1
2
=
�
�

54. PROBABILITY OF
Independent events
3.If a coin and a die are tossed, what is the
probability of getting a headand an odd
number?
??????(�)getting a head (coin)
�
�
??????(�)getting an odd number (die)
3
6
or
�
�
??????�∩�=
1
2
∙
1
2
=
�
�

55. PROBABILITY OF
Independent events
4.The probability that Rodrigo will pass the
exam is
3
5
. The probability that Antonio will
pass the same exam is
5
6
. If each of them takes
the exam, what is the probability that:
a.bothRodrigo andAntonio will pass.
b.Rodrigo will pass andAntonio will fail.
c.Antonio will pass and Rodrigo will fail.
d.one of the twowill pass. (Rodrigo or Antonio)
e.bothRodrigo and Antonio will fail.

56. a. bothRodrigo andAntonio will pass.
3
5
∙
5
6
=
15
30
=
�
�
b. Rodrigo will pass andAntonio will fail.
3
5
∙
1
6
=
3
30
=
�
��
c. Antonio will passand Rodrigo will fail.
5
6
∙
2
5
=
10
30
=
�
�
d. one of the twowill pass. (Rodrigo or Antonio)
1
10
+
1
3
=
��
��
e. bothRodrigo andAntonio will fail.
2
5
∙
1
6
=
2
30
=
�
��

57. PROBABILITY OF
Independent events
5.A large box contains 3 green balls and 6 red
balls. Two balls are drawn at random from the
box, one at a time and with replacement. What
is the probability that the first ball drawn is
greenand the second ball drawn is red?
??????(�)drawing a green ball (1
st
ball)
3
9
or
�
�
(�) drawing a red ball (2
nd
ball)
6
9
or
�
�
??????�∩�=
1
3
∙
2
3
=
�
�

58. PROBABILITY OF
dependent events
6.A large box contains 3 green balls and 6 red
balls. Two balls are drawn at random from the
box, one at a time and without replacement.
What is the probability that the first ball
drawn is greenand the second ball drawn is
red?
??????(�)drawing a green ball (1
st
ball)
3
9
or
�
�
P(�) drawing a red ball (2
nd
ball)
6
8
or
�
�
??????�∩�=
1
3
∙
3
4
=
3
12
=
�
�

60. Conditional probability
•Conditional probabilityis the probability that
an event will occur given that another event has
already occurred.
•If �and �are any events, then
??????��=
??????(�??????�??????�)
??????(�)
??????��=
??????(�∩�)
??????(�)
??????��is read as “the probability of A given B”.

61. Conditional probability
1.The probability that Leni studies and passes
her summative test in Math is
3
5
. If the
probability that she studies is
9
10
, what is the
probability that she passes her summative test,
giventhat she studied?
??????(�) passes (not given)
??????(�) studies
9
10
??????(�∩�)studies and passes
3
5
??????��=
3
5
9
10
=
30
45
=
�
�

62. Conditional probability
2.When two dice are rolled, what is the
probability that the sum of the numbers
appeared is 7ifit is known that one of the
numbers is 3?
??????(�) sum of the numbers is 7
6
36
or
�
�
??????(�) one of the numbers is 3
��
��
??????(�∩�)sum is 7 & one number is 3
2
36
or
�
��
??????��=
1
18
11
36
=
36
198
=
�
��

63. Conditional probability
3.In rolling two dice, what is the probability that
both numbers appeared are evengiventhat the
product of the two numbers is 12?
??????(�) both numbers are even
9
36
or
�
�
??????(�) product is 12
4
36
or
�
�
??????(�∩�)both are even & product is 12
2
36
or
�
��
??????��=
1
18
1
9
=
9
18
=
�
�

64. Conditional probability
4.A card is drawn from a deck of 52 cards. What
is the probability that it is a black queenifit is
known that the card is a spade?
??????(�) black queen
2
52
or
�
��
??????(�) spade
13
52
or
�
�
??????(�∩�)both black queen and spade
�
��
??????��=
1
52
1
4
=
4
52
=
�
��

65. Conditional probability
5.In a bowl, there are 20 chips numbered from 1
to 20. If a chip is drawn, what is the probability
that the chip shows an even numbergiventhat
it is a composite number?
??????(�) even number
10
20
or
�
�
??????(�) composite number
��
��
??????(�∩�)both even and composite
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9
20
11
20
=
180
220
=
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66. Here are simple explanations for item
numbers 2 to 5 on how to obtain the
conditional probability in an easy way.

67. Conditional probability
2.When two dice are rolled, what is the
probability that the sum of the numbers
appeared is 7ifit is known that one of the
numbers is 3?
There are 36 outcomes (36 pairs of numbers) if
two dice are rolled. 11 of them have a 3 in one of
the two numbers{(3,1), (3,2), (3,3), (3,4), (3,5),
(3,6), (1,3), (2,3), (4,3), (5,3), (6,3)}.Out of these
11 outcomes, 2 have a sum of 7{(3,4), (4,3)}.
Therefore, ??????��=
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68. Conditional probability
3.In rolling two dice, what is the probability that
both numbers appeared are evengiventhat the
product of the two numbers is 12?
There are 36 outcomes (pairs of numbers) when
two dice are rolled. There are 4 outcomes where
the product of the two numbers is 12{(2,6), (6,2),
(3,4), (4,3)}.Out of these 4 outcomes,there are 2
where both numbers are even{(2,6), (6,2)}. So,
??????��=
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=
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69. Conditional probability
4.A card is drawn from a deck of 52 cards. What
is the probability that it is a black queenifit is
known that the card is a spade?
From a deck of 52 cards, there are 13 spades. Out
of these 13, there is 1 black queen.
Hence, ??????��=
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70. Conditional probability
5.In a bowl, there are 20 chips numbered from 1
to 20. If a chip is drawn, what is the probability
that the chip shows an even numbergiventhat
it is a composite number?
From 1 to 20, there are 11 composite numbers
{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20}.Out of these
11 composite numbers, 9 are even numbers{4,
6, 8, 10, 12, 14, 16, 18, 20}. Thus, ??????��=
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