 # Use Exhaustive Events

In this worksheet, students practise identifying exhaustive events and look at the probability of something not happening. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Probability

Curriculum subtopic:   Probability, Combined Events and Probability Diagrams

Difficulty level:   ### QUESTION 1 of 10

We have looked so far at finding a single probability. This time, we're going to look at exhaustive events.

What are exhaustive events?

Exhaustive events are two events that describe everything that could happen.

For example, If I throw a coin, getting a head and getting a tail are the only two things that could happen so they are exhaustive

If a roll a dice, getting a 2 and getting a 3 are not exhaustive as they don't describe everything that could happen

If I roll a dice, getting a 2 and not getting a 2 are exhaustive as they describe everything that could happen.

We know that if something is certain, it has a probability of 1.

We can use this fact to find the probability of finding the probability of something not happening.

Example: Find the probability of rolling a dice and not getting a 2

Let's look at the example from before with the dice.

The probability I get a 2 and the probability I don't get a 2 are exhaustive, this means one of them is certain to happen.  This means we can say that...

P(getting a 2) + P(not getting a 2) = 1

If we rearrange this...

P(not getting a 2) = 1 - P(getting a 2).

This means that if we know the probability of something happening, we can find the probability of it not happening.

P(not getting a 2) = 1 - 1/6 = 5/6

If I have two events that describe every possible outcome, we call these...

I know the probability of option A is 3/5.

What is the probability of not getting option A?

I roll a biased dice that has the probability of getting a six as 1/3

What is the probability of not getting a six?

True or False

Two probabilities will always add to 1.

True

False

Match the probabilities for these exhaustive events.

## Column B

Option A = 1/2
Option B = 3/10
Option A = 2/3
Option B = 1/2
Option A = 7/10
Option B = 7/9
Option A = 2/9
Option B = 1/3

The probability  it will rain tomorrow is 0.55.

What is the probability it won't rain?

If I pick a letter at random from a sentence, the probability the letter is a G is 3/8.

What is the probability is won't be a G.

Select if these events are exhaustive or not.

The probability a runner completes a marathon in under 3 hours of 72%.

What is the probability he takes more than 3 hours?

Exhaustive events are events that describe...

• Question 1

If I have two events that describe every possible outcome, we call these...

EDDIE SAYS
In this, we say we have 'exhausted' all the things that could happen. They are exhaustive events.
• Question 2

I know the probability of option A is 3/5.

What is the probability of not getting option A?

2/5
EDDIE SAYS
We know that A and not A are exhaustive so they add to 1. This means the probability of not getting A is 1 - the probability of A 1 - 3/5 = 2/5
• Question 3

I roll a biased dice that has the probability of getting a six as 1/3

What is the probability of not getting a six?

2/3
EDDIE SAYS
We know that getting a six and not getting a six are exhaustive so they add to 1. This means the probability of not getting a six is 1 - the probability of getting a six 1 - 1/3 = 2/3
• Question 4

True or False

Two probabilities will always add to 1.

False
EDDIE SAYS
They can, but this doesn't mean they will. This will only be true if the events are mutually exclusive
• Question 5

Match the probabilities for these exhaustive events.

## Column B

Option A = 1/2
Option B = 1/2
Option A = 2/3
Option B = 1/3
Option A = 7/10
Option B = 3/10
Option A = 2/9
Option B = 7/9
EDDIE SAYS
Because you are told these are exhaustive events, all you need to be doing is looking for the ones that add up to a whole. i.e. 2/3 +1/3 = 1
• Question 6

The probability  it will rain tomorrow is 0.55.

What is the probability it won't rain?

0.45
EDDIE SAYS
We know that probabilities can be given as fractions, decimals or percentages. This does just follow the same rules as before. Raining and not raining are exhaustive so their probabilities must add up to 1. 1 - 0.55 =
• Question 7

If I pick a letter at random from a sentence, the probability the letter is a G is 3/8.

What is the probability is won't be a G.

5/8
EDDIE SAYS
Exhaustive events again peeps. 1 - 3/8 = ?
• Question 8

Select if these events are exhaustive or not.

EDDIE SAYS
Remember that things are exhaustive when they describe all the things that could happen. A 5 and a 6 on a dice aren't exhaustive because you could also get a 1,2,3 or 6
• Question 9

The probability a runner completes a marathon in under 3 hours of 72%.

What is the probability he takes more than 3 hours?

EDDIE SAYS
You need to think a bot more carefully here. If we write probabilities as percentages, we aren't looking for ones that add to 1, but ones that add to 100%
• Question 10

Exhaustive events are events that describe...

EDDIE SAYS
I can\'t stress this enough. If these describe all the possible outcomes, they are exhaustive events.
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