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Conditional probability using Venn diagrams
Conditional probability is the probability of an event occurring given that another event has occurred.
● the probability of David studying GCSE mathematics given that he is studying GCSE physics,
● the probability that I will pay my gas bill given that I have just been paid,
● the probability that my students will turn up to class given that it is a rainy day.
The emphasis is that the probability is influenced by something that has already happened.
P(A|B) means the probability of A occurring, given that B has already occurred.
📌 Ex1. The diagram shows the number of students in a year group who are female (set A) and the number of left handed students in the same year group (set B).
(a) Write down the P(A∩B).
(b) Write down the P(A∪B).
(c) How many students are in the year group altogether?
(d) A student is chosen at random from the year group. What is the probability that the student is a right—handed given that the student is female?
📌 Ex2. In a class of 29 girls
13 girls play hockey
7 girls play both
(a) Show this information on a Venn diagram.
(b) One of these girls is picked at random. Write down the probability that this girl plays hockey given that this girl also plays netball.
N∩H= This is how many play both.
N∩H̄=N–H=18 play netball but 7 play both so need another 11 who play netball.
N̄∩H=H–N= 13 play hockey but 7 play both so need another 6 who play hockey.
(N∪H)=This shows how many do not play netball or hockey so 29-(11+7+6)=5.
(b) 18 girls play netball ←[Given plays netball so look at the netball circle only]
7 of these the netball players also play hockey ←[We want to know how many of these netball players also play hockey]
Probability plays hockey given plays netball =7/18
📌 Ex3. In a group of 40 children there are 19 who can swim and 16 who can ride a bike. There are 5 children who can swim and ride a bike.
(a) A child is selected at random. Find the probability that this child cannot swim or ride a bike.
(b) Another child is selected at random. Given that this child can ride a bike, work out the probability that this child can swim.
(a). It is helpful to draw a Venn diagram to show this information.
S∩B= This shows how many can swim and ride a bike.
S∩B̄=S–B= 19 can swim but 5 of these can already swim so need another 14 in the circle for swimming.
S̄∩B=B–S= 16 can ride a bike but 5 of these can already ride a bike so need another 11 in the circle for bike.
(S∪B)=This shows how many cannot swim or ride a bike so 40-(14+5+11)=10.
🔑 Answer: P(S∪B)=10/40=¼=0.25 ←[10 of the 40 children cannot swim or ride a bike]
NOTE: You do not need to cancel the answer to ¼.
(b) 5+11=16 ride a bike ←[Given rides a bike so look at the bike circle only]
5 of these bike riders also swim ←[We want to know how many of these bike riders also swim]
Probability that a child can swim given rides a bike =5/16.
📌 Ex4. Of 36 people, 17 have an interest in reading magazines and 12 have an interest in reading books, 6 have an interest in reading both magazines and books.
a) Represent the information in a Venn diagram.
Let B and M be the people who read books and magazines respectively.
b) How many people have no interest in reading magazines or books?
11 people only read magazines, 6 people only read books and 6 people read both.
The total number of people is 36. Therefore we can find the number of people who have no interest in reading magazines or books:
c) If a person is chosen at random from the group, find the probability that the person will:
|i. have an interest in reading magazines and books.
ii. have an interest in reading books only.
iii. not have any interest in reading books.
📌 Ex5. In a survey at Lwandani Secondary School, 80 people were questioned to find out how many read the Sowetan, how many read the Daily Sun and how many read both. The survey revealed that 45 read the Daily Sun, 30 read the Sowetan and 10 read neither. Use a Venn diagram to find the percentage of people that read:
a) only the Daily Sun
The following Venn diagram represents the given information. However we can calculate more information from this that will help us answer the problem.
We note the following information:
● 45 people read the Daily Sun, some of these also read the Sowetan.
● 30 people read the Sowetan, some of these also read the Daily Sun.
● Of the total number of people questioned 10 did not read either newspaper, so 80-10=70 read neither.
Let the number of people who read the Daily Sun only be d, the number of people who read the Sowetan only be s and the number of people who read both be x. Now we note the following:
We can fill this in on the Venn diagram:
b) If a person is chosen at random from the survey, find the probability that the person will have an interest in reading only the Sowetan
c) If a person is chosen at random from the survey, find the probability that the person will have an interest in reading both the Daily Sun and the Sowetan
📌 Ex6. Anjali asked 60 students in her year group about where they had eaten out in the last month. Here are her results:
|26 had eaten in Subfood
11 had eaten in Macdinner and Subfood
12 had not eaten at Macdinner or Subfood
(a) Draw a Venn diagram to represent this information.
(b) Find the probability of a student who had eaten at Macdinner.
📌 Ex7. A running club has 120 members.
|88 of the members take part in road races
55 of the members take part in marathons
17 of the members do not run in road races or in marathons
Work out the probability that a member only takes part in a road race or in a marathon but not both.
📌 Ex8. In a survey 100 people were asked whether they watched snooker or cricket when it was on TV. 20 watched neither, 75 watched snooker, 32 watched cricket.
A person is selected at random.
(a) Find the probability that this person watched both cricket and snooker.
(b) Given that this person watched snooker, work out the probability that this person watched cricket.