1. Lecture 9 - Probability
C2 Foundation Mathematics (Standard Track)
Dr Linda Stringer Dr Simon Craik
l.stringer@uea.ac.uk s.craik@uea.ac.uk
INTO City/UEA London

2. Probability
Probability can be expressed as a percentage, a fraction or
a decimal. In this course we usually express probability as
a fraction.
The probability that an event A occurs is a fraction between
0 and 1. It is written P(A).
P(A) =
The number of ways that A can occur
The total number of possible outcomes
The sum of the probabilities of all events is always equal
to 1.

3. Probability
P(A) =
The number of ways that A can occur
The total number of possible outcomes
Question: I roll a fair die. What is the probability that I roll
a 2?
Answer: There is one way of rolling a 2, and there are 6
possible outcomes. P(2) = 1
6 .
The event A is ’I roll a 2’, and we write P(2) for P(I roll a 2).
Question: I roll a fair die. What is the probability that I roll
a 5?
Answer: There is one way of rolling a 5, and there are 6
possible outcomes. P(5) = 1
6 .

4. Probability
P(A) =
The number of ways that A can occur
The total number of possible outcomes
Question: I roll a fair die. What is the probability that I roll
an odd number?
Answer: There are three ways that I can roll an odd
number (if I throw a 1, or 3, or 5), and there are six
possible outcomes. P(odd) = 3
6 = 1
2 .
Question: I roll a fair die. What is the probability that I roll a
number less than three?
Answer: There are two ways that I can roll a number less
than three (if I throw a 1 or 2), and there are six possible
outcomes. P(less than 3) = 2
6 = 1
3.

5. Not (¬)
The probability that A does not happen is 1 minus the
probability that it does.
P(¬A) = 1 − P(A).
Example: What is the probability that I do not roll a 5?
Answer: P(¬5) = 1 − P(5) = 1 − 1
6 = 5
6 .

6. Or
The probability that an event A or an event B happens is
the sum of the probability of A and the probability of B.
P(A or B) = P(A) + P(B).
(In this course we consider mutually exclusive events only)
Question: I roll a die once. What is the probability that I roll
a 4 or a 5?
Answer: P(5 or 6) = P(5) + P(6) = 1
6 + 1
6 = 2
6 = 1
3 .

7. And
The probability that an event A and an event B happens is
the probability of A multiplied by the probability of B.
P(A and B) = P(A) × P(B).
(In this course we consider independent events only)
Question: I roll a die twice. What is the probability that I roll
a 4 both times?
Answer: P(4, 4) = P(4) × P(4) = 1
6 × 1
6 = 1
36 .
Question: I roll a die twice. What is the probability that I roll
a 4 the first time and a 5 the second time?
Answer: P(4, 5) = P(4) × P(5) = 1
6 × 1
6 = 1
36 .

8. And and Or
Question: I roll a die twice. What is the probability that I roll
a four and a five (in any order)?
Answer: P(4, 5 or 5, 4) = P(4, 5) + P(5, 4) =
(1
6 × 1
6 ) + (1
6 × 1
6) = 1
36 + 1
36 = 2
36 = 1
18.
Question: I roll a die twice. What is the probability that I roll
at least one four?
Answer:
P(4, 4 or 4, ¬4 or ¬4, 4) = P(4, 4) + P(4, ¬4) + P(¬4, 4) =
(1
6 × 1
6 ) + (1
6 × 5
6) + (5
6 × 1
6 ) = 1
36 + 5
36 + 5
36 = 11
36 .
This is the same as
1 − P(¬4, ¬4) = 1 − (5
6 × 5
6 ) = 1 − 25
36 = 11
36

9. Probability trees
A probability tree is a method of representing more than
one event
Draw different branches for different events at each stage,
and write the probability next to the branch.
Multiply probabilities along the branches (AND)
Add up the probabilities at the end of each branch (OR)
The total of all the probabilities at the end of the branches
should be 1
The probabilities for the second stage may be the same as
for the first stage, or they may be different

10. Probability tree - tossing two coins (or tossing one coin
twice)
Coin 1 Coin 2
P(H, H) = 1
2 × 1
2 = 1
4
P(H, T) = 1
2 × 1
2 = 1
4
P(T, H) = 1
2 × 1
2 = 1
4
P(T, T) = 1
2 × 1
2 = 1
4
T
T
1/2
H1/2
1/2
H
T
1/2
H1/2
1/2
What is the probability that I get two heads? (1
4 )
What is the probability that I get exactly one tail? (1
2 )
What is the probability that I get at least one head? (3
4 )
What is the probability that I get no heads? (1
4)

11. Probability tree - roll a fair die twice (or roll two fair
dice)
First roll Second roll
P(4, 4) = 1
6 × 1
6 = 1
36
P(4, ¬4) = 1
6 × 5
6 = 5
36
P(¬4, 4) = 5
6 × 1
6 = 5
36
P(¬4, ¬4) = 5
6 × 5
6 = 25
36
¬4
¬4
5/6
41/6
5/6
4
¬4
5/6
41/6
1/6
What is the probability that I roll two fours? ( 1
36)
What is the probability that I roll no fours? (25
36 )
What is the probability that I roll exactly one four? (10
36)
What is the probability that I roll at least one four? (11
36)

12. Probability tree - shoot a ball at a basket twice
p(GOAL)=0.7
First shot Second shot
NO GOAL
NO GOAL
0.3
GOAL0.7
0.3
GOAL
NO GOAL
0.3
GOAL0.7
0.7
What is the probability of two goals?
What is the probability of no goals?
What is the probability of exactly one goal?
What is the probability of ten goals

13. Probability socks
I have 10 pairs of socks. Four pairs are black, three pairs
are red and three pairs are green. They are all jumbled up
in a drawer. I take out two socks (without replacement)
What is the probability that they are both red?
What is the probability that I pick a matching pair?

14. Probability socks
There are 20 socks in the drawer. We take one out and do not
replace it, so when we take the second sock there are only 19
socks in the drawer. Therefore the probabilities on the second
tier are different from the probabilities on the first tier.
6
20
8
20
6
20
Black
Red
Green
7
19
6
19
6
19
Black
Red
Green
6
19
8
19
5
19
Black
Red
Green
6
19
8
19
5
19
Black
Red
Green