Extending Probability
Mutually Exclusive Events

If two events cannot happen or take place at the same time, then they are called mutually exclusive events. For example when tossing a single coin the events 'heads' and 'tails' are mutually exclusive because they cannot both be obtained at the same time.

If A and B are mutually exclusive events then the probability of obtaining A or B is given by:

p(A or B) = p(A) + p(B)

Worked Examples

1

State whether or not the pairs of events described below are mutually exclusive.

(a)

A: A die is rolled and shows a 6.

B: A die is rolled and shows an odd number.

Show me

It is not possible for a die to show a six and an odd number at the same time, so these events are mutually exclusive.
(b)

A: Selecting a child with blue eyes from a class.

B: Selecting a left handed person from a class.

Show me

It is possible to select a person who has blue eyes and is left handed, so these events are not mutually exclusive.
2

When Andrew buys a can of drink the probabilities of selecting particular brands are given in the table below:

DrinkProbability
Cola
Lemonade
Fizzo

Find the probabilities that he selects:

(a)

Cola or Lemonade

Show me
p(Cola or Lemonade) = p(Cola) + p(Lemonade)
= + = +
=
(b)

Fizzo or Lemonade

Show me
p(Fizzo or Lemonade) = p(Fizzo) + p(Lemonade)
= +
= +
=
(c)

none of the drinks listed above.

Show me

First find the probability that he chooses one of the drinks from the list.

p(Cola or Lemonade or Fizzo) = p(Cola) + p(Lemonade) + p(Fizzo)
= + +
= + +
=

then

p(drink not from list) = 1 − p(drink from list)
= 1 −
=

Exercises

When a die is rolled, the following outcomes can be used to describe the result.

A: An odd number is obtained.

B: An even number is obtained.

C: A prime number is obtained.

Complete the following statements.

A and are mutually exclusive.

B and are not mutually exclusive.

Decide if each pair of events A and B given below are mutually exclusive.

(a)

A: Winning a football match.

B: Losing a football match.
(b)

A: Selecting a diamond from a pack of playing cards.

B: Selecting a King from a pack of playing cards.
(c)

A: Arriving late on a train journey.

B: Not arriving early on a train.
(d)

A: Selecting an ace from a pack of cards.

B: Selecting a queen from a pack of cards.
(e)

A: It rains tomorrow.

B: It is sunny tomorrow.

When Plymouth Argyle football team play in a league match the probability that they win is 0.4 and the probability that they draw is 0.3. What is the probability that they lose?

When a car arrives at a set of traffic lights the probability that a green light is showing is and the probability that a red light is showing is .

What is the probability that neither the green or red lights are showing?

A bag contains 6 red counters, 5 blue counters and 4 pink counters. A counter is selected from the bag at random.

Find, the probability that the counter is:

(a)
either red or pink,
(b)
not pink,
(c)
not red,
(d)
blue or pink.

A bag contains a number of balls of different colours. The probability of obtaining a ball of a particular colour is given in the table below.

ColourProbability
Red
Green
Blue

What is the probability that a ball taken from the bag is:

(a)
red or green,
(b)
not blue or green,
(c)
not one of the colours listed above?

In a class of 25 students there are 8 students with size 6 feet, 4 students with size 7 feet and 3 students that are left handed.

Find:

(a)

the probability that a student in this class has size 6 or size 7 feet,

(b)

the probability that a student in this class is right-handed.

Is it possible to find the probability that:

(c)
a student has size 6 feet and is left-handed,
(d)
has feet that are bigger than size 7,
(e)
has feet that are not size 6 or size 7?

The spinner shown is biased.

The probabilities of getting a particular colour are shown in the table below:

(a)

Complete the table to show the probability of getting Green.

ColourRedYellowBlueGreen
Probability0.40.10.3
(b)

The spinner is spun once. What is the probability of getting either Red or Blue?

(c)

The spinner is spun 50 times. Approximately how many times would you expect to get Red?

A bag contains a total of 20 beads. There are 6 red beads, 9 blue beads and 5 white beads.

(a)

A bead is taken at random from the bag. The probability that it is red is 0.3.

(i)
What is the probability that it is white?
(ii)
What is the probability that it is not white.

All the beads are taken from the bag, numbered 1, 2, 3 or 4 and then replaced.

When a bead is taken from the bag at random the probability of each number is as shown in the table.

Number
on bead		Probability
10.3
20.4
30.2
40.1

The red beads are numbered,

(b)

A bead is taken at random from the bag.

What is the probability that it is red or numbered 4?