math probability

1. Probability
Digital Lesson

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Any activity with an unpredictable results is called an
experiment.
The results of an experiment are called outcomes and the
set of all possible outcomes is the sample space.
Examples: Identify the sample space.
Flip a coin.
Toss a die.
S = {H, T}
S = {1, 2, 3, 4, 5, 6}
The number of outcomes in the sample space S is n(S).
2
6
Experiment Sample Space n(S)

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Any subset of the sample space is called an event.
Examples: List the outcomes in each event.
Flip a coin
Toss a die
Toss a die
Draw a card
Flip two coins
The number of outcomes in an event E is n (E).
3
4
4
3
Get heads {H}
Get an even number {2, 4, 6}
Get a 3 or higher {3, 4, 5, 6}
Get an 8 {8, 8, 8, 8}
Get at least one head {HH, HT, TH}
Experiment Event n(E)
1

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If n(E) = 0, then P(E) = 0, and the event is impossible.
If E is an event from a sample space S of equally likely
outcomes, the probability of event E is:
If n(E) = n(S), then P(E) = 1 and the event is certain.
( )
( )
( )
n E
P E
n S

Examples: A 6-sided die is rolled once.
The event is impossible.
The event is certain.
P(10) = = 0
6
0
P(n  10) = = 1
6
6
P(5) =
6
1
Note that 0  P(E)  1.

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Example 1: Two coins are tossed. What is the probability that at
least one head comes up?
S = {HH, HT, TH, TT} E = {HH, HT, TH}
( ) 3
( )
( ) 4
n E
P E
n S
 
Example 2: A card is drawn at random from a standard deck of
52 cards. What is the probability the card drawn is a
face card?
S = all 52 cards in the deck n(S) = 52
E = {J, J, J, J, Q, Q, Q, Q, K, K, K, K}
n(E) = 12
13
3
52
12
)
(
)
(
)
( 


S
n
E
n
E
P

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Two events A and B are mutually exclusive if they have
no outcomes in common, A  B = .
Example: When a die is tossed, which events are mutually
exclusive?
A: getting an even number C: getting 5 or 6.
B: getting an odd number
The Venn diagram shows that only A  B = , therefore,
only events A and B are mutually exclusive.
C
A
2
4
6
B
1
3
5

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J
J
J
B
K
Q
A
J
A  B
If A and B are events, their union A  B, is the event “A or B”
consisting of all outcomes in A or in B or in both A and B.
A  B = {J, J, J, J, Q, K }
Example: A card is drawn at random from a standard
deck of 52 cards.
A: getting a club face card B: getting a jack.
List the outcomes for the event of getting a club face card or
getting a jack.

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If A and B are events, their intersection, written A  B, is the event “A
and B” consisting of all outcomes common to both A and B.
Example: A card is drawn at random from a standard deck of 52
cards.
A  B = {J}
A: getting a club face card
List the outcomes for the event of getting a club face card and
getting a jack.
B: getting a jack.
J
K
Q
A
J
J
J
B
A  B

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(T, H)
(H, T)
S
(T, T)
(H, H)
A
(T, H)
(H, T)
If A is an event, the complement of A, written A , is the event “not A”
consisting of all outcomes not in A.
Examples: Two coins are flipped.
= {(H, H), (T, T)}
A
List the outcomes for the event not getting one head and one tail?
A
Event A is getting one head and one tail.

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B
).
(
)
(
)
(
)
( B
A
P
B
P
A
P
B
A
P 




If A and B are events, the probability of “A or B” is:
A
– n(A  B)
+ +
A  B
= ( + ) + ( + ) –
n(A  B) = n(A) + n(B)

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).
(
)
(
)
( B
P
A
P
B
A
P 


If A and B are mutually exclusive, then
A B
n(A  B)
+
A and B are mutually exclusive
A  B = 0
= n(A) + n(B)
= +

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52
26
)
red
( 
P
52
2
)
queen
red
a
is
card
the
(
)
queen
red
( 

 P
P









52
28
52
2
52
4
52
26
)
queen
red
(
)
queen
(
)
red
( P
P
P
)
queen
red
(
)
queen
a
or
red
is
card
the
( 
 P
P
13
7
Example: A card is drawn at random from a standard deck of
52 cards. What is the probability the card is red or a queen?
“queen”
Q
Q
52
4
)
queen
( 
P
7
“red”
J
K
Q
5
J
9
8
4
6
10
Q
6
7
2
A
9
K
4
10
3
5
8
2
3

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Example 2: A card is drawn at random from a standard
deck of 52 cards.What is the probability the
card is a spade or a club?
Since these events are mutually exclusive,
P(club or spade) = P(club) + P(spade) = .


4
1
4
1
2
1
 J
 6
 7
2
 A
 9
 K
 4
 10
 3
 5
8
Q
“spade” “club”
Q
 J
 6
 7
 2
 A
 9
 K
 4
 10
 3
 5
 8

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For example, when flipping two coins, the events
“the first coin comes up heads” and
“the second coin comes up tails” are independent.
Two events are independent if the fact that one event
has occurred has no effect on likelihood of the other
event.
( ) ( ) ( )
P A B P A P B
  
If A and B are independent events, the probability of “A
and B” is:

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Example: A card is drawn at random from a standard
deck of 52 cards. What is the probability the
card is a red queen?
B: the card is a queen
Events A and B are independent.
26 4 1
( and ) ( ) ( ) ( )
52 52 26
P A B P A B P A P B
      
A: the card is red

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Example: A die is tossed. What is the probability of
getting 2 or higher?
It is easier to work with the complementary event
“getting a 1”which has probability .
1
6
If A is an event, the probability of the event “not A” is:
)
(
1
)
( A
P
A
P 


6
5
6
1
1
)
higher
or
two
a
getting
(
)
one
a
getting
not
( 


 P
P