Lesson Objective
Understand the Basic Terminology used in probability
Understand the use of the symbols ∩ and U and their rough
translations in to everyday English
What is Probability?
Probability is about measuring the likelihood that an event happens as
part of a trial
A trial is an experiment that has a number of possible outcomes
The possibility space is a list of all the possible outcomes connected with
a trial
An event is a subset of the possibility space consisting of the outcomes
we are interested in
If all the outcomes of a trial are equally likely then:
P(event happens) =
Number of ways an event can happen
Number of items in the possibility space
In a survey of 200 people, 75 said they had shopped at Tesco in the last week,
whilst 123 said that they had shopped at Sainsburys. 17 had not visited either
supermarket.
If you picked a shopper at random from the 200 in the survey, what is the
probability:
a)That they had not visited either Sainsburys or Tescos?
b)That they had visited Tescos?
c)That they had not visited Sainsburys?
d)That they has shopped in at least one of Tescos or Sainsburys?
A
B
A
B
A
B
A
B
Venn Diagrams are Great For Probability
Eg Suppose there are 100 pupils in a school sixth form, 50 study maths, 29
studying biology and 13 studying both
Draw a Venn Diagram and calculate:
a) P(B’)
b) P(M ∩ B)
c) P(M ∩ B’)
d) P(M U B)
e) P(M U B’)
f) P(M’ ∩ B)
g) What can you say about P(B) and P(B’) – is this generally true?
Venn Diagrams are Great For Probability
Eg Suppose there are 100 pupils in a school sixth form, 50 study
29 studying biology and 13 studying both
M
B
37
13
16
34
maths,
Lesson Objective
Consolidate the use of the symbols ∩ and U and their rough
translations in to everyday English
Begin to consider rules for calculating P(A U B)
I roll 2 fair six sided die. One is Red, the other is Blue.
The word total refers to the sum of the score on the two dice.
Let R = Score on the Red Die is Even
B = Score on the Blue Die is less than 3
T = Total score is 7
W = Total score is Even
Find:
a)
b)
c)
d)
e)
f)
g)
h)
P(R)
P(R ∩ B)
P(R U B)
P(T ∩ R)
P(B )
P(B ∩ T)
P(R ∩ W)
P(R U W)
i) P(T U W)
j) P(W ∩ B)
k) P(B’ U R)
l) P(T’ ∩ W)
m) P(T’ U W’)
n) P(B U T’)
o) P( (R U B) ∩ T)
p) P( (W ∩ R) U T)
I roll 2 fair six sided die. One is Red, the other is Blue.
The word total refers to the sum of the score on the two dice.
Let R = Score on the Red Die is Even
B = Score on the Blue Die is less than 3
T = Total score is 7
W = Total score is Even
Find:
a)
b)
c)
d)
e)
f)
g)
h)
P(R)
P(R ∩ B)
P(R U B)
P(T ∩ R)
P(B )
P(B ∩ T)
P(R ∩ W)
P(R U W)
½
6/36
24/36
3/36
1/3
2/36
9/36
27/36
i) P(T U W)
j) P(W ∩ B)
k) P(B’ U R)
l) P(T’ ∩ W)
m) P(T’ U W’)
n) P(B U T’)
o) P( (R U B) ∩ T)
p) P( (W ∩ R) U T)
24/36
6/36
30/36
18/36
1
32/36
4/36
15/36
Lesson Objective
Understand what it means for two events to be mutually exclusive
Begin to consider rules for calculating P(A U B) and understand
how events being mutually exclusive affects these rules
If two events are Mutually Exclusive what does this mean?
Give me an example of two events that are mutually exclusive
If two events are Mutually Exclusive what does this mean?
Give me an example of two events that are mutually exclusive
A
B
A
B
Mutually exclusive means that the two events do not overlap.
They cannot happen at the same time or, in maths notation:
P(A ∩ B) = 0
I roll a red die and a black die. Draw a possibility space diagram to
show the total scores for the two dice.
a) P(A total of 4 U a total of 6)
b) P(A total of 8 U a total that is prime)
c) P(A total that is even U a total more than 9)
d) P(A 4 on the red die U a 6 on the black die)
e) P(A total that is even U a total that is prime)
f) P(A total that is more than 7 U a 5 on the black die)
g) P(A total less than 5 U a total more than 10)
It is often said at GCSE that:
P(A or B) = P(A U B) = P(A) + P(B)
When is this rule true, how can we modify it to make a rule that is
true all the time?
I roll a red die and a black die.
Find each probability and decide if the events are mutually exclusive:
a) P(A total of 4 U a total of 6)
= 8/36 ME
b) P(A total of 8 U a total that is prime)
= 20/36 ME
c) P(A total that is even U a total more than 9)
= 20/36
d) P(A 4 on the red die U a 6 on the black die) = 11/36
e) P(A total that is even U a total that is prime) = 32/36
f) P(A total that is more than 7 U a 5 on the black die) = 17/36
g) P(A total less than 5 U a total more than 10) =9/36 ME
At GCSE we often say that P(A OR B) = P(A) + P(B)
What we mean by this is that P(A U B) = P(A) + P(B)
iff A and B are mutually exclusive
What we should really say is that:
P(A OR B) = P(A) + P(B) – P(overlap)
Or using proper mathematical notation:
P( A  B)  P( A)  P( B)  P( A  B)
1) In a group of 20 adults, 4 out of the 7 women and 2 out of the 13 men wear glasses.
What is the probability that a person chosen at random from the group is a woman
or someone who wears glasses?
2) In a street containing 20 houses, 3 households do not own a television set, 12
households have a black and white set, and 7 households have a colour and a black
and white set. Find the probability that a household chosen at random owns a colour
television set.
2
1
1
3) Given that , P(A’) = , P(B) = , P(A ∩ B) =
. Find P(A U B)
3
2
12
4) A card is selected at random from a regular set of 52 playing cards. Let Q be the
event that the card is a queen and D be the event that the card is a diamond. Draw a
Venn diagram to show these events.
Find
a) P(Q ∩ D)
c) P(Q’ U D)
b) P(Q U D)
d) P(Q ∩ D’)
1) In a group of 20 adults, 4 out of the 7 women and 2 out of the 13 men wear glasses.
What is the probability that a person chosen at random from the group is a woman
or someone who wears glasses? 9/20
2) In a street containing 20 houses, 3 households do not own a television set, 12
households have a black and white set, and 7 households have a colour and a black
and white set. Find the probability that a household chosen at random owns a colour
television set. 12/20
2
1
1
3) Given that , P(A’) = , P(B) = , P(A ∩ B) =
. Find P(A U B) 9/12
3
2
12
4) A card is selected at random from a regular set of 52 playing cards. Let Q be the
event that the card is a queen and D be the event that the card is a diamond. Draw a
Venn diagram to show these events.
Find
a) P(Q ∩ D) 1/52
c) P(Q’ U D) 49/52
b) P(Q U D)
d) P(Q ∩ D’)
16/52
3/52
Lesson Objective
Derive a rule to find the P(A ∩ B)
Understand when two events are independent and be able and the
affect that independent events has on the rule
In maths it is often said that P(A ∩ B) = P(A) × P(B)
Consider the following situations and decide if this rule is true:
I roll a fair Blue die and a fair Red die
R = Score a Red 6
M = Red Score more than 3
B = Score a Blue 6
N = Blue Score more than 4
Find
P(R)
P(R ∩ B)
P(R ∩ M)
P(B)
P(M ∩ N)
P(R ∩ N)
Is it true that:
P(R ∩ B) = P(R) × P(B)
P(M ∩ N) = P(M) × P(N)
P(B ∩ N) = P(B) × P(N)
P(B ∩ M) = P(B) × P(M)
P(R ∩ M) = P(R) × P(M)
P(R ∩ N) = P(R) × P(N)
P(N)
P(B ∩ N)
P(M)
P(B ∩ M)
In maths it is often said that P(A ∩ B) = P(A) × P(B)
Consider the following situations and decide if this rule is true:
I roll a fair Blue die and a fair Red die
R = Score a Red 6
M = Red Score more than 3
B = Score a Blue 6
N = Blue Score more than 4
Find
P(R) 1/6
P(B) 1/6
P(M) 3/6
P(N) 2/6
P(R ∩ B) 1/36 P(M ∩ N) 6/36 P(B ∩ N) 6/36 P(B ∩ M) 3/36
P(R ∩ M) 6/36 P(R ∩ N) 2/36
Is it true that:
P(R ∩ B) = P(R) × P(B)
P(M ∩ N) = P(M) × P(N)
P(B ∩ N) = P(B) × P(N)
P(B ∩ M) = P(B) × P(M)
P(R ∩ M) = P(R) × P(M)
P(R ∩ N) = P(R) × P(N)
yes
yes
no
yes
no
yes
Why does the rule
work sometimes
and not others?
What stops it from
working all the
time?
The problem is independence!
The rule P(A ∩ B) = P(A) × P(B) only works if the events A and B
are events that are independent
P(Blue 6)
P(Blue score more than 4)
These events not independent because if I know that the score is a
Blue 6 then the Blue score must be bigger than 4
If knowing the result of one outcome alters the probability of the
second outcome then we say that the items are not independent
Independent or not?
1) I flip a fair coin twice.
A = Head on 1st flip
B = Head on 2nd flip
2) A draw has 3 blue and 3 yellow socks. I draw two socks from the a
draw at the same time
A = 1st sock yellow
B = 2nd sock yellow
3) I pick 2 different students from this class.
A = 1st student a girl
B = 2nd student a Boy
4) I look at the weather on the 1st day and last days of a 10 day holiday
A = It snows on day 1st day
B = It snows on 2nd day
5) I pick a teacher and student from a school
A = Teacher’s name is David
B = Student’s name is David
In fact the rule should be:
P(A ∩ B) = P(A) × P(B‫׀‬A)
This is called a conditional probability and means:
The probability that ‘B happens given that we know
A has already happened’
If A and B are independent then P(B‫׀‬A) = P(B)
And the rule reverts back to P(A ∩ B) = P(A) × P(B)
Lesson Objective
Be able to calculate conditional probabilities by counting
Begin to use conditional probabilities with our rules two
probability rules:
P(A U B) = P(A) + P(B) – P(A U B)
P(A ∩ B) = P(A) x P(B ‫ ׀‬A)
to solve problems:
Consider rolling two fair die – one red and one blue:
R = 6 on the Red die
B = 6 on the Black die
T = Total score of 12
E = Total score is Even
Find: a) P(R ‫ ׀‬B)
c) P(R ‫ ׀‬T)
e) P(R ‫ ׀‬E)
g) P(B ‫ ׀‬T)
i) P(T ‫ ׀‬E)
k) P(R’ ‫ ׀‬B)
m)P(T’ ‫ ׀‬E)
b) P(B ‫ ׀‬R)
d) P(T ‫ ׀‬R)
f) P(E ‫ ׀‬R)
h) P(T ‫ ׀‬B)
j) P(E ‫ ׀‬T)
l) P(B’ ‫ ׀‬R)
b) P(E’ ‫ ׀‬T)
a) Is it true that P(A ‫ ׀‬B) = P(B ‫ ׀‬A) ?
b) What do you notice about P(A ‫ ׀‬B) and P(A’ ‫ ׀‬B )?
Use puzzle pieces on the next two pages with this table to
complete the puzzle:
Lesson Objective
Be able to use tree diagrams to solve probability questions
Focus in particular on questions that include a ‘reversed conditional
probability’ part (Bayes theorem)
Consider the following probability question that may (or may not be)
answered with a tree diagram:
The probability that it is sunny on any given day is 1/3. If it is sunny the
probability that I go swimming is 3/4 , if it is not sunny the probability
that I go swimming is only 1/5.
a) Draw a tree diagram to illustrate this information
b) What is the probability that I go swimming?
c) If you know that I went swimming what is the probability that
it was sunny on that particular day?
Consider the following probability question that may (or may not be) answered
with a tree diagram:
The probability that it is sunny on any given day is 1/3. If it is sunny the
probability that I go swimming is 3/4 , if it is not sunny the probability that I go
swimming is only 1/5.
a) Draw a tree diagram to illustrate this information
b) What is the probability that I go swimming?
c)
If you know that I went swimming what is the probability that
it was sunny on that particular day?
Note:
To answer these ‘backwards’ conditional probability questions it is
important to notice that:
P(A ∩ B) = P(B ∩ A)
So that P(A) × P(B ‫׀‬A) = P(A ∩ B) = P(B ∩ A) = P(B) × P(A ‫׀‬B)
Meaning that P(A ‫׀‬B) = P(A ∩ B)
P(B)
(Essentially the number of times that A happens at the same time as
B as a proportion of the number of times B happens)
3. A student regularly has to take two train journeys on a Sunday. Over a long period of time
she has worked out the probability that the first train is late is 0.4. If the first train is late,
the probability that the second train is late is 0.5. If the first train is not late, the
probability that the second train is late is 0.3.
(i) Draw a tree diagram to show the possible outcomes for her next 2 journeys on a Sunday.
(ii) Using the tree diagram calculate the probability:
(A) that exactly one journey is on time,
(B) that both trains are on time, given that the second journey is on time.
5. In the build up to the Olympics a high jumper measured her success at a particular height.
She has a maximum of 3 attempts at this height; once she has jumped successfully she
does not jump that height again. On 60% of occasions she clears it at the first attempt.
When she attempts the height for the second time, she is successful in 75% of the
attempts. When she attempts the height for the third time, she is successful in only 30%
of the attempts.
(i) Draw a tree diagram to show the possible outcomes.
(ii) From the tree diagram calculate the probability
(A) that she successfully clears the height in her three attempts,
(B) that, given she clears the height, that she does so successfully at the first attempt.
Lesson Objective
Understand the seriousness of misunderstanding the difference
between P(A ‫׀‬B) and P(B ‫׀‬A). Recognise that people do this all the
time in every day life and it is very easy to slip into
Imagine that there is a rare, but deadly, disease that only affects 0.1%
of the population. The test for the disease is very reliable, being
accurate 95% of the time.
You decide to test for the disease and the result comes back positive.
How worried should you really be?
Imagine that there is a rare, but deadly, disease that only affects 0.1%
of the population. The test for the disease is very reliable, being
accurate 95% of the time.
You decide to test for the disease and the result comes back positive.
How worried should you really be?
Test positive given
that you have disease
0.95
Have disease 0.001
Don’t have disease
0.999
P(have disease ‫ ׀‬+ve) = P(+ve ∩have disease)
P(+ve)
= 0.00095 ÷ (0.00095+0.04995)
= 0.0187 ie around 2% chance !!!
Have Dis and +ve =
0.001 x 0.95 = 0.00095
Test negative given
that you have disease
0.05
Have Dis and -ve =
0.001 x 0.05 = 0.00005
Test positive given
that you do not have
disease 0.05
Don’t have Dis and
+ve = 0.999 x 0.05 =
0.04995
Test negative given
that you do not have
disease 0.95
Don’t have Dis and ve = 0.999 x 0.95 =
0.94905
Lesson Objective:
Final Probability Overview Practise
Look at the variety of different probability questions
A school has 120 students in year 12.
94 students do English or Maths
57 students do maths and 49 do English
a) What is the probability that a student chosen at random
does neither Maths or English
b) What is the probability that a student chosen at random
does both English and Maths
c) What is the probability that a student chosen at random
does English but not Maths
d) If two students are selected at random what is the
probability that they both do Maths and English
Number of
Xmas
presents
bought by
sixth form
students
Probability
for Mrs
Smith
Probability
for Mr
Banham
0
0.6
0.4
1
0.2
0.3
2
0.1
0.1
3
0.1
0.2
a)
What is the probability
that Mrs Smith receives at
least one present?
b)
What is the probability
that Mr Banham and Mrs
Smith both receive no
presents?
c)
What is the probability
that Mr Banham receives
more presents than Mrs
Smith?
d)
Given that Mr Banham
received more presents
than Mrs Smith what is
the probability that he
received 3 presents.
There are two A-level maths classes in Year 12. Group X contains eleven girls and
seven boys. Group Y contains six girls and eight boys.
Two students are chosen at random. Find the probability that
a) they are of the same gender,
b) they are from the same group,
c) they are of the same gender and from the same group,
d) they are of the same gender, given that they are from the same group,
e) they are from the same group, given that they are of the same gender.
In a large Sixth Form, 40% of the students do Mathematics, 30% do Physics and 24%
do both Mathematics and Physics.
a) Illustrate this information using a Venn diagram.
b) Hence or otherwise find the probability that a student, chosen at random:
(i) does Physics but not Mathematics,
(ii) does neither Mathematics nor Physics,
(iii) does not do Physics, given that he or she does not do Mathematics.