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Introduction
Cards
Combining Probabilities
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Introduction
• Probability is the maths of chance and gambling,
telling us how likely an event is to occur.
• The probability of an event occurring is usually
written as a fraction or as a percentage.
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• Toss one coin, assume it is a fair coin and ignore
the chance of it landing on its edge.
• There are only two possible outcomes –
either a head H or a tail T.
H
P(Head) =
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T
1
__
2
P(Tail) =
1
__
2
Definition
The probability of an event occurring is:
Number of times this event occurs
P (event) = –––––––––––––––––––––––––––––
Total possible number of outcomes
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 Cards 
• Consider a deck of cards. There are four suits called hearts,
diamonds, clubs and spades.
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 Cards 
• Consider a deck of cards. There are four suits called hearts,
diamonds, clubs and spades.
• In each of these there are 13 cards – an ace, the numbers 2 to
10 inclusive, and the picture cards: jack, queen and king.
• This makes a total of 52 cards in an ordinary deck. Some
games require one extra card, the Joker. We will not use this
extra card.
• If the cards are boxed (shuffled or mixed) and then 13 cards
dealt to each of four people, the chances of a particular person
getting 13 clubs are 635,013,559,600 to 1.
• This is more than the number of seconds in 20,000 years!
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 Cards 
Clubs 
A
2
3
4
5
6
7
8
9 10 J
Q K
Diamonds  A
2
3
4
5
6
7
8
9 10 J
Q K
Hearts 
A
2
3
4
5
6
7
8
9 10 J
Q K
Spades 
A
2
3
4
5
6
7
8
9 10 J
Q K
P(Clubs) =
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13
___
52
number of clubs in deck
1
__
=
total number of cards
4
 Cards 
Clubs 
A
2
3
4
5
6
7
8
9 10 J
Q K
Diamonds  A
2
3
4
5
6
7
8
9 10 J
Q K
Hearts 
A
2
3
4
5
6
7
8
9 10 J
Q K
Spades 
A
2
3
4
5
6
7
8
9 10 J
Q K
P(2) =
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4
___
52
number of 2s in deck
total number of cards
=
1
__
13
Combining Probabilities
• If two events happen simultaneously, a sample space
can be constructed to see clearly the possible
outcomes.
• A sample space involves putting all the possible
outcomes of one event on one axis of a grid and all
the possible outcomes of a second event on the other
axis.
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Dice
Throw 1
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1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
P(both same) =
6
___
36
number the same
total outcomes
Throw 2
36
=
1
__
6
Dice
Throw 1
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1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
P(two 4s) =
1
___
36
Throw 2
36
number the same __
1
=
total outcomes
36
Dice
Throw 1
Quit
1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
P(at least one 6) =
11
___
36
Throw 2
36
Dice
Throw 1
Quit
1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
P(not getting a 6) =
25
___
36
Throw 2
36
Dice
Throw 1
Quit
1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
3
___
1
__
P(a total of 10) =
36
=
12
Throw 2
36
Three or more events
P(2 heads and 1 tail) =
H
T
Quit
3
___
H
HHH
8
T
HHT
H
HTH
T
T
HTT
H
H
THH
T
THT
H
TTH
T
TTT
H
T
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