Lesson 9: �Using frequencies and probabilities
Objectives
1
Medical testing
2
Starla
Toby
Starla and Toby are developing medical tests for �a disease.
A blood test gives a definite result (‘Yes’ or ‘No’).
How well does a self-administered test work?
Lab trial
3
Starla
Toby
FALSE POSITIVE
FALSE NEGATIVE
How might you feel if you got�a false positive?
Don’t have disease.
Test positive.
Have disease.
Test negative.
Lab trial results table
4
No. of participants | | Total | Test result | Total | |
20 | Have disease | | Test positive (+) | | |
Test negative (−) | | | |||
Do not have disease | | Test positive (+) | | | |
Test negative (−) | | | |||
Don’t have disease.
Test positive.
Don’t have disease.
Test negative.
Have disease.
Test negative.
Have disease.
Test positive.
Lab trial results
5
No. of participants | | Total | Test result | Total | |
20 | Have disease | | Test positive (+) | | |
Test Negative (−) | | | |||
Do not have disease | | Test positive (+) | | | |
Test negative (−) | | | |||
Is this a�good test?
Don’t have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test positive.
Have disease.
Test negative.
Don’t have disease.
Test positive.
Don’t have disease.
Test negative.
Don’t have disease.
Test negative.
Don’t have disease.
Test negative.
Don’t have disease.
Test negative.
Don’t have disease.
Test negative.
Don’t have disease.
Test negative.
Don’t have disease.
Test negative.
Representing the results
6
have the
disease
don’t have�the disease
+ test
− test
+ test
− test
How could we represent the results?
No. of participants | | Total | Test result | Total | |
20 | Have disease | | Test positive (+) | | |
Test Negative (−) | | | |||
Do not have disease | | Test positive (+) | | | |
Test negative (−) | | | |||
11
9
7
2
1
10
Official trial results
7
have the
disease
don’t have�the disease
+ test
1001
− test
+ test
− test
202
151
51
398
401
DISCUSS
Add up to �1001
Add up to �202
Add up to �799
The numbers
at the ends of arms
from the same starting point add up to the starting point value.
799
Probability tree diagram
8
have the
disease
don’t have�the disease
+ test
1001
− test
+ test
− test
202
151
51
799
401
398
have the
disease
don’t have
the disease
+ test
+ test
− test
− test
___
___
___
0.2
a
e
d
c
b
Why can 0.2 be used rather than 0.202?
YOUR TURN
Probability model
9
have the
disease
don’t have�the disease
+ test
1001
− test
+ test
− test
202
151
51
799
401
398
have the
disease
don’t have
the disease
+ test
+ test
− test
− test
___
___
___
0.2
___
0.8
0.5
0.5
0.25
0.75
REVIEW
Add up to 1
Add up to 1
Add up�to 1
Why is this diagram wrong?
10
have the
disease
don’t have
the disease
+ test
+ test
− test
− test
0.2
The probabilities on each pair of branches should add to one.
DISCUSS
have the
disease
don’t have�the disease
+ test
1001
- test
+ test
- test
202
151
51
398
401
799
Is this diagram wrong?
11
have the
disease
don’t have
the disease
+ test
+ test
− test
− test
0.2
The rounding we use should give a good approximation.
DISCUSS
have the
disease
don’t have�the disease
+ test
1001
- test
+ test
- test
202
151
51
398
401
799
Scaling up: fill in the empty ovals
12
DISCUSS
+ test
10 000
− test
+ test
− test
2000
1500
500
8000
4000
4000
have the
disease
don’t have�the disease
have the
disease
don’t have
the disease
+ test
+ test
− test
− test
___
___
___
0.2
0.8
0.75
0.5
0.5
0.25
Using probability to scale up
13
YOUR TURN
Handout�available
Row A: discussion
14
REVIEW
Add up to 1
Add up to 1
have the
disease
+ test
+ test
− test
− test
___
0.5
___
0.8
___
___
___
___
0.2
0.5
0.1
0.9
Why is this statement incorrect?
5000 people are likely to have the disease
Add up to 1
don’t have
the disease
Row B: an incorrect diagram
15
REVIEW
have the
disease
don’t have
the disease
+ test
+ test
− test
− test
Which values on the probability tree are not correct?
Hockey injuries
16
DISCUSS
injury >�concussion
___
0.4
___
___
___
___
___
Winter season
Summer season
injury >
no concussion
injury >�concussion
injury >�concussion
Injury >
no concussion
injury >
no concussion
The probability of a hockey injury that results in concussion in any one season is 0.4.
0.4
0.4
0.6
0.6
0.6
Calculating probabilities
17
DISCUSS
injury >�concussion
___
0.4
___
___
___
___
___
Winter season
Summer season
injury >
no concussion
0.4 × 0.4 = 0.16
injury >�concussion
injury >�concussion
Injury >
no concussion
injury >
no concussion
What is the probability of a player �having an injury resulting in concussion in both seasons?
0.4
0.4
0.6
0.6
0.6
Do we add�or multiply along the branches?
Using probabilities
18
DISCUSS
50 of the members get injured
15 000 of the students get injured
How many people would you expect to have two concussion injuries from playing hockey during the year for the following?
0.16 × 50 = 8
0.16 × 15 000 = 2400
A hockey club with 75 members:
18 000 students play hockey across all the universities in the UK:
Using probabilities
19
YOUR TURN
Handout�available
Row C: correct diagram
20
REVIEW
injury >�broken bone
___
___
___
___
___
___
injury >�no break
injury >�broken bone
injury >�broken bone
injury >�no break
injury >�no break
0.15
0.85
0.85
0.85
0.15
0.15
0.15 × 0.15 = 0.0225
We would expect 1800 �of the 80 000 injured registered players �to sustain two injuries involving a broken bone during the year.
0.0225 × 80 000 = 1800
Row C: incorrect values
21
REVIEW
injury >�broken bone
injury >�no break
injury >�broken bone
injury >�broken bone
injury >�no break
injury >�no break
What mistake has been made?
Row D: working with fractions
22
REVIEW
injury >�sprain
___
___
___
___
___
___
injury >�no sprain
injury >�sprain
injury >�sprain
injury >�no sprain
injury >�no sprain
× =
We would expect 400 of the 900 injured students to have two injuries involving a sprain during the year.
× 900 = 400
True or False?
23
Statement 2:
Statement 1:
Statement 3:
Statement 5:
REVIEW
Statement 4:
Practice question (1)
24
REVIEW
Handout�available
Each worker in a factory is either left-handed or right-handed.
22 of the 45 workers are male.
16 of the 34 right-handed workers are female.
Complete the frequency tree for this information.
(3)
Q14 from June 2018, 1MA1/2F
22
23
16
18
4
7
male
female
left-handed
left-handed
right-handed
right-handed
45
Practice question (2)
25
REVIEW
Handout�available
When a biased 6-sided dice is thrown once, the probability that it will land on 4 is 0.65.
The biased dice is thrown twice.
Amir draws this probability tree diagram.
The diagram is not correct.
Write down two things that are wrong with the probability tree diagram.
1 The probabilities should total 1, so the 0.25 should be 0.35.
2 The 0.35 and 0.65 in the first branches for the second throw are the wrong way around.
(2)
Q22 from June 2018, 1MA1/3F
land on 4
land on 4
land on 4
not land on 4
not land on 4
not land on 4
0.65
0.65
0.65
0.25
0.35
0.35
first throw
second throw
Lesson review: �Using frequencies and probabilities
26
Suggested further steps/areas to work on
Objectives
Lesson 9: �Credits
Text acknowledgements�Pearson Edexcel Level 1/Level 2 GCSE (9-1) Mathematics Paper 2 (Calculator) June 2018 Foundation Tier Paper Reference 1MA1/2F, Pearson Edexcel Level 1/Level 2 GCSE (9-1) Mathematics Paper 3 (Calculator) June 2018 Foundation Tier Paper Reference 1MA1/3F
27