Probability
• The probability of an event occurring is between 0 and 1
• If an event is certain not to happen, the probability is 0
eg: the probability of getting a 7 when you roll a die = 0
• If an event is sure to happen, the probability is 1
eg: the probability of getting either a head or a tail when you
flip a coin = 1
• All other events have a probability between 0 and 1
Probability
Impossible
Very unlikely
to happen
Not likely
to happen
Likely
to happen
Equal chance
of happening
Likely or unlikely?
Certain
Very Likely
to happen
Relative Frequency
This gives information about how often an event occurred
compared with other events.
eg: Maths Exam results from 26 students
Exam %
> 80
No. of students Relative Frequency
10
10
26 = 0.38 (2 dp)
60 - 80
12
12
= 0.46 (2dp)
26
40 - 60
3
< 40
1
3
= 0.12 (2 dp)
26
1
= 0.04 (2 dp)
26
Sample Space
The set of all possible outcomes is called the sample space.
eg. If a die is rolled the sample space is:{ 1, 2, 3, 4, 5, 6 }
eg. If a coin is flipped the sample space is:{ H, T }
eg. For a 2 child family the sample space is: { BB, BG, GB, GG }
Equally likely outcomes
Probability = Number of favourable outcomes
Number of possible outcomes
In many situations we can assume outcomes are equally likely.
When events are equally likely:
Equally likely outcomes
may come from, for
example: experiments
with coins, dice,
spinners and packs of
cards
Favourable Outcomes are the results we want
Possible Outcomes are all the results that are possible - the
SAMPLE SPACE
Examples:
1
Pr (getting a 5 when rolling a dice) =
6
3 1

Pr (even number on a dice) =
6 2
4
16
Pr (J, Q, K or Ace in a pack of cards) =
= 13
52
Lattice Diagrams
A spinner with numbers 1 to 4 is spun and an unbiased coin
is tossed. Graph the sample space and use it to give the
probabilities:
(a) P(head and a 4)
1
P(head and a 4) =
8
P(head or a 4) = 5
8
(those shaded)
(b) P(head or a 4)
Sample space
A spinner with numbers 1 to 4 is spun and
an unbiased coin is tossed.
We can work out the sample space by a lattice diagram or a
tree diagram.
Lattice Diagram
Sample Space is {H1, H2, H3, H4, T1, T2, T3, T4}
Lattice Diagrams
The sample space when rolling 2 dice can be shown by
the following lattice diagram:
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)
1
6
Pr (double) =
=
6
36
Pr (total ≥ 7) =
21
7
=
36
12
Tree Diagrams for Probability
A tree diagram is a useful way to work out probabilities.
eg: Show the possible combination of genders in a 3 child family
Pr (2 girls & a boy) =
1st child
B
G
3
8
3rd child
2nd child
B
Outcomes
BBB
B
G
B
G
G
B
BBG
BGB
G
B
BGG 
GBB
G
B
GBG 
GGB 
G
GGG
Experimental Probability
When we estimate a probability based on an experiment, we call the
probability by the term “relative frequency”.
Relative frequency =
num berof successful outcom es
total num berof trials
The larger the number of trials, the closer the experimental probability
(relative frequency) is to the theoretical probability.
Probability
Relative Frequency
Theoretical
Experimental
Relating factor “is the term we use in”
Favourable Outcomes are the results we want
Possible Outcomes are all the results that are possible – (the
sample space)
Examples:
Pr (getting a 5 when rolling a dice) =
Pr (even number on a dice) =
Pr (J, Q, K or Ace in a pack of cards) =
1
6
3 1

6 2
16
4
=
52
13
Complementary Events
When rolling a die, ‘getting a 5’ and ‘not getting a 5’ are complementary
events. Their probabilities add up to 1.
Pr (getting a 5) = 1
Pr (not getting a 5) = 5
6
6
Using Grids (Lattice Diagrams) to find Probabilities
6
● ● ● ● ● ●
5
● ● ● ● ● ●
4
● ● ● ● ● ●
3
● ● ● ● ● ●
2
● ● ● ● ● ●
1
● ● ● ● ● ●
red die
1
2
3
Rolling 2 dice:
6
Pr (double) =
36
1
=
6
21
7
Pr (total ≥ 7) =
=
36
12
4 5 6
blue die
Rolling a die & Flipping a coin:
T
● ● ● ● ● ●
coin H
● ● ● ● ● ●
Pr (tail and a 5) =
Pr (tail or a 5) =
1
2
3
4
die
5
6
1
12
7
12
Using Grids (Lattice Diagrams) to find Probabilities
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)
1
6
Pr (double) =
=
6
36
Pr (total ≥ 7) =
21
7
=
36
12
Multiplying Probabilities
In the previous lattice diagram, when rolling a die and flipping a coin,
T
● ● ● ● ● ●
coin H
● ● ● ● ● ●
1
2
3
4
die
5
6
Pr (tail) =
1
2
Pr (tail and a 5) =
Pr (5) =
1 x 1
2
6
=
1
6
1
12
So Pr (A and B) = Pr (A) x Pr (B)
example: Jo has probability ¾ of hitting a target, and Ann has
probability of ⅓ of hitting a target. If they both fire
simultaneously at the target, what is the probability that:
a) they both hit it
b) they both miss it
ie Pr (Jo hits and Ann hits)
ie Pr (Jo misses and Ann misses)
= Pr (Jo hits) x Pr (Ann hits)
ie Pr (Jo misses) x Pr (Ann misses)
=¾x⅓
=¼x ⅔
1
=¼
=
6
Tree Diagrams to find Probabilities
In the above example about Jo and Ann hitting targets, we can
work out the probabilities using a tree diagram.
Let H = hit, and M = miss
Ann’s results Outcomes
H and H
H
Jo’s results ⅓
¾
H
¼
M
⅔
⅓
⅔
Pr (both hit) = ¼
Probability
¾x⅓=¼
M
H
H and M
M and H
¾x⅔=½ ●
or
¼ x ⅓ = 121 ●
M
M and M
¼x⅔=
total = 1
1
Pr (both miss) = 6
Pr (only one hits) ie Pr (Jo or Ann hits) = ½ + 1 =
12
1
6
7
12
Expectation
When flipping a coin the probability of getting a head is ½, therefore
if we flip the coin 100 times we expect to get a head 50 times.
Expected Number = probability of an event occurring x the number of trials
eg: Each time a rugby player kicks for goal he has a ¾ chance of being
successful. If, in a particular game, he has 12 kicks for goal, how many
goals would you expect him to kick?
Solution: Pr (goal) = ¾
Number of trials = 12
Expected number = ¾ x 12 = 9 goals