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What is the probability of winning the first prize in Lotto?
Why to people toss a coin to make a decision?
What other tools do we use in a game of chance?
Say whether each of these events is ‘certain’, ‘likely’,
‘unlikely’ or impossible to occur.
a.) You will live in the same house for the rest of your life
b.) You will toss a die and roll a ‘six’
c.) The sun will set in the west tonight
d.) It will be colder where in February than in August
e.) The mail will be delivered tomorrow
f.) It will rain next month
List all the possible totals
you can roll with two
normal dice in a table.
What is the probability of
obtaining:
4 1
=
a.) a total of 5
36 9
1
2
b.) a total of 11 36 = 18
c.) a ‘two’ on the black die,
and a ‘six’ on the white die.
1
2
3
4
5
6
1 2
2 3
3 4
4 5
5 6
6
7
7
8
3
4
4
5
5
6
6
7
5
6
6
7
8
7
8 9
7 8 9 10
8 9 10 11
9 10 11 12
How many possible
outcomes?
1
36
36
List all the possible totals
you can roll with two
normal dice in a table.
Roll two dice 50 times and
record the total of the
numbers on the dice
in a table.
Outcome
2
11
12
Tally
Frequency
1
2
3
4
5
6
1 2
2 3
3 4
4 5
5 6
6
7
7
8
3
4
4
5
5
6
6
7
5
6
6
7
8
7
8 9
7 8 9 10
8 9 10 11
9 10 11 12
1.) From your table what is
your experimental
probability of rolling:
a.) a six?
b.) a one?
c.) a total less than 4?
d.) an even number?
e.) not an even number?
f.) What happens when
you add your answers
from part d and e ?
g.) Is 1 a possible outcome?
Outcome
Tally
Frequency
2
1
3
4
5
6
3
4
8
8
7
8
9
8
7
4
10
3
11
12
2
2
If a trial has ‘n’ equally likely outcomes, and
a success can occur ‘s’ ways, then the
probability of a success is:
P(success) =
s
n
e.g. What is the probability of tossing a ‘heads’
n=2
Flipping a coin has 2 equally likely outcomes
Tossing a head is a success, this can only
occur 1 way
P(heads) =
1
2
If a trial has ‘n’ equally likely outcomes, and
a success can occur ‘s’ ways, then the
probability of a success is:
P(success) =
s
n
This scale shows how we can describe the probability of an event
0
0.5
1
Probabilities can be written as fractions, decimals or
percentages.
Probabilities are always between 0 and 1.
The sample space is a list of all possible outcomes.
The probability of all possible outcomes always add to 1.
IGCSE Ex 6
Pg 329-331
A jar contains a large number of marbles coloured
red, green, yellow, orange and blue. A marble was
chosen at random, its colour noted and then
replaced. This experiment was carried out 200 times.
Here are the results.
Colour
Frequency
Red
Green
Yellow
Orange
Blue
52
34
38
44
32
What is the probability that a randomly selected marble is:
a.) orange
44
11
=
200
50
b.) green
34
= 17
200 100
c.) not green
d.) red or blue
166
= 83
200 100
52  32
21
=
200
50
Experimental probability from an experiment
repeated a large number of times can be useful
to make predictions about events.
Toss a coin 100 times.
Record how many times it lands on heads in a table.
After every 10 throws calculate the fraction of heads so far.
Convert your proportions (fractions) to decimals.
Graph the number of throws vs. the proportion of heads.
# of # of heads Proportion
As a
Tosses (frequency)
of heads decimal
(relative freq)
Graph the
number of
tosses vs. the
proportion of
heads
What do you
notice about
the proportion
of heads
tossed as the
number of
tosses
increases?
10
/10
20
/20
30
/30
40
/40
50
/50
60
/60
70
/70
80
/80
90
/90
100
/100
Long run proportions can be obtained by
repeating the experiment a number of times
• there will always be some variation in experiments
because chance is involved
• probability becomes more accurate as more trials are
carried out (closer to theoretical probability)
When outcomes of an event are equally likely,
their probabilities are the same.
If A is a particular event then:
P(A) =
number of outcomes in A
total number of possible outcomes
P(A) means ‘the probability that A will occur’
The compliment (opposite of A) is all the possible
outcomes not in A and is written A’ (not A)
BETA Ex 33.03
P(not A) = 1 – P(A)
Pg 946-949
Two events are exclusive if they cannot occur at the
same time
e.g.
Rolling a die and having it be an even number and
rolling a ‘3’.
e.g.
Drawing from a pack of cards a black card and a diamond
For exclusive events, A and B
P(A or B) = P(A) + P(B)
e.g. A marble is selected from a bag containing
3 red, 2 white, and 5 purple. What is the
probability of selecting a red OR a white ball?
These are exclusive events
P(R or W) = P(R) + P(W)
3
2
=
+
10
10
5
= 10
= 1
2
Two events are independent if the occurrence of one
does not affect the other.
e.g. Rolling a die and tossing a coin at the same time.
For independent events, A and B
P(A and B) = P(A) × P(B)
e.g. A fair die and a coin are tossed. What is the probability of
obtaining a ‘tails’ and an even number on the die?
These are independent events
P(Tails and even) = P(tails) × P(even)
1
= 2 x
1
=
2
1
4
e.g.
Two cards are drawn from a pack of 52, one after the other.
The first card is replaced before the second card is drawn.
What is the probability that both cards are Aces?
P(2 Aces) = P(Ace) x P(Ace)
1
1
x
13 13
1
=
169
The first card is not replaced before the second card is drawn. What
is the probability that both cards are Aces?
=
e.g.
P(2 Aces no replacement) = P(Ace) x P(Ace)
3
4
= 52 x 51 =
1
221
1
.
e.g. The probability that it will rain on any day in May is
4
Find the probability that:
a.) it will rain on both May the 1st and May the 21st.
1
1
1
=
x
=
4
4
16
b.) it will not rain on May the 21st.
P(not rain) = 1 – P(rain)
1
=1- 4
3
= 4
c.) it will rain on May the 1st, but not on May the 21st.
1
3
3
P(rain and not rain) =
x
=
4
4
16
IGCSE Ex 7
Pg 332
If we know the probability of an event, we can
predict roughly how often the event will occur.
Expected Number = Number of trials x Probability of event
e.g.
How many times would we expect a ‘three’ to
occur when a fair die is rolled 120 times.
P(three) =
1
6
Number of trials = 120
Expected number of ‘threes’ = 120 x 1 = 20
6
Expected Number = Number of trials x Probability of event
e.g. When playing basketball the probability of getting a
basket from inside the key is 0.75. If you make 20 shots, how
many can you expect to go in?
P(basket) = 0.75
Number of trials = 20
Expected number of ‘baskets’ = 20 x 0.75 = 15
Expected Number = Number of trials x Probability of event
e.g.
The percentage of students that pass an examination
is 45%. If 700 students sit the examination, how many
students would be expected to pass?
Number of students = 700 x 0.45
= 315
BETA Ex 33.04
Pg 953
Tree diagrams are useful for listing outcomes of
experiments that have 2 or more successive events
(choices are repeated)
• the first event is at the end of the first branch
• the second event is at the end of the second branch etc.
• the outcomes for the combined events are listed on the
right-hand side.
The probability of some events can also be
found using a probability tree.
Each branch represents
a possible outcome.
A node is a point where
a choice is made.
Node
e.g.
The possibilities when a couple have 2
children are:
B
1
2
1
2
1
2
e.g. P(B,B) =
B
1
2
1
2
G
1
2
G
B
1
2
x
1
2
G
1
=4
Every possible outcome
must be represented by
a branch from a node
The sum of the
probabilities on the
branches from each
node is 1
To calculate the
probabilities of a
sequence of events,
we multiply the
probabilities along
the branches.
R
SR
H
SH
How many
possible results
are there?
2x3=6
Find (random select)
P(single) =
D
V
DV
R
H
DR
DH
1
2
P(single vanilla)=
P(raspberry)=
1
6
2 1
=
6 3
BETA Ex 30.05
Pg 962
Draw a tree diagram to show some
alternative ways you could spend your
Saturday
You must first do a chore (choose
from 4 options),
Then you can choose to either
watch a DVD or visit friends (2
options)
4 x 2 = 8 possible outcomes
A bag contains 5 red balls and 3 green
balls. A ball is selected and then
replaced. A second ball is selected.
Find the probability of selecting:
R
a.) Two green balls
3
8
x
3
8
=
9
64
3
8
5
8
3
8
5
8
5
8
G
3
8
R
G
R
G
A bag contains 5 red balls and 3 green
balls. A ball is selected (NOT replaced). A
second ball is selected.
Find the probability of selecting:
R
a.) Two green balls
3
8
x
2
7
=
6
56
=
3
28
b.) One red and one green
5
8
x
3
7
+
3
8
x
5
7
3
7
5
8
3
8
=
30
56
=
15
28
4
7
5
7
G
2
7
R
G
R
G
IGCSE Ex 8
Pg 333-336
On a Monday or a Thursday, Mr. Picasso
paints a ‘masterpiece’ with a probability
of 1 . On any other day, the probability of
5
1
producing a masterpiece is 100 . Mr.
Picasso never knows what day it is, so
what is the probability that on a random
day he will produce a masterpiece?
There are 7 days in the week
2
7
5
7
there is a probability of
there is a probability of
2
7
x
1
5
+
5
7
x
1
100
=
9
140
1
5
1
100
A venn diagram presents information in groups.
Set A can be written:
A = {a, b, c, e}
The number of elements in set A is 4, written n(A) = 4
Set B = {c, d, e, f, h} and n(B) = 5
The rectangle ε represents the
universal set
ε = {a, b, c, d, e, f, g, h} and n(ε) = 8
The overlap of the two set represents the intersection
of the sets.
A ∩ B = {c, e}
n(A ∩ B) = 2
The union is the set of elements in A
or B or in both sets.
A U B = {a, b, c, d, e, f, h}
n(A U B) = 7
A’ is the complement of the set A. It contains all the
elements of ε which are not in A.
A’ = {d, f, g, h}
n(A’) = 4
(A U B)’ = {g}
(A ∩ B)’ = {a, b, d, f, h, g}
A only = {a, b}
e.g. In a year 10 class of 25 students, 18 enjoy
watching basketball and 15 enjoy watching
tennis. If a student is chosen at random,
find the probability that he:
a.) enjoys watching both basketball and tennis
n(B) = 18
n(ε) = 25
n(T) = 15
n(B ∩ T) = 8
8
P(B ∩ T) =
25
b.) enjoys watching tennis only
7
P( T only) =
25
BB
10
Tennis
8
7