Download 8-7: Conditional Probability

Name_________________________________
Date: ____________
Lesson 8-7: Conditional Probability
Learning Goals:
#8: What is conditional probability? How do we determine a conditional
probability?
Conditional Probability
In words: The probability of an event (A), given that another event (B) has already occurred.
In symbols: 𝑃( 𝐴| 𝐵) =

𝑃(𝐴 and 𝐵)
𝑃(𝐵)
(given in formula booklet)
Restrict the total outcomes to be the outcomes of B. Then find the probability of both events
occurring.
Example: In a school of 1200 students, 250 are seniors, 150 students take math, and 40 students are
seniors and are also taking math.
What is the probability that a randomly chosen student is taking math, given they are a senior?
Let’s try an example using information from our class:
P(Boy|Boots)=
Male
Boots
P(Girl|Boots)=
No Boots
Total
P(Boots|Girl)=
Female
Total
Finding Conditional Probability
2
2
1) Given that 𝑃(𝐶) = 3 𝑎𝑛𝑑 𝑃(𝐶 ∩ 𝐷) = 5 , 𝑓𝑖𝑛𝑑 𝑃(𝐷|𝐶).
2) At a school, 60% of students buy a school lunch. Only 10% of students buy lunch and dessert. What is
the probability that a student who buys lunch also buys dessert?
3) Let P(A) = 0.6, P(B) = 0.5 and 𝑃(𝐴 ∩ 𝐵) = 0.2
a. Draw a Venn diagram to represent the given information
b. Find each of the following probabilities:
i. 𝑃(𝐴 ∪ 𝐵)
ii. 𝑃((𝐴 ∪ 𝐵)′ )
iii. 𝑃(𝐴|𝐵)
iv.𝑃(𝐵|𝐴)
4) Given that P(B) = .8 and P(A ∩ 𝐵) = 0.6, 𝑓𝑖𝑛𝑑 𝑃(𝐴|𝐵).
5) The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a
week, the probability that it is Friday is 0.2. What is the probability that a student is absent given that
today is Friday?
Finding Conditional Probability from Tables
6) Researchers analyzed 200 adults to determine if there was
a link between their highest level of education and
whether or not they smoked. Use the table at the right to
answer the following questions, leave all questions in
fraction form and use proper notation.
a) Given the person smoked, what's the probability they
graduated high school?
b) Given the person smoked, what's the probability they went to a 2 year college?
7) In a survey, 100 students were asked “do you prefer to watch television or play sports?” Of the 46
boys in the survey, 33 said they would choose sports, while 29 girls made this choice.
Boys
Girls
Total
Television
13
25
38
Sport
33
29
62
Total
46
54
100
By completing this table or otherwise, find the probability that
a) a student selected at random prefers to watch television;
b) a student prefers to watch television, given that the student is a boy.
Finding Conditional Probability from Tree Diagrams
8) The events B and C are dependent, where C is the event “a student takes Chemistry”, and B is the
event “a student takes Biology”. It is known that the probability a student takes Chemistry is 0.4. For
students that take Chemistry the probability they also take Biology is 0.6, otherwise the probability
they take Biology is 0.5.
a) Complete the following tree diagram.
Chemistry
Biology
Remember:
B
0.4
C
B
In a tree diagram you
must multiply along
the branches to
determine
probabilities!
B
C
B
b) Calculate the probability that a student takes Biology.
c) Given that a student takes Biology, what is the probability that the student takes Chemistry?
9) A teacher has a box containing six type A calculators and four type B calculators. The probability that
a type A calculator is faulty is 0.1 and the probability that a type B calculator is faulty is 0.12.
a) Complete the tree diagram given below, showing all the probabilities.
0.1
FAULTY
type A
0.6
NOT FAULTY
FAULTY
0.4
type B
NOT FAULTY
b) Given you pick a not faulty calculator, what’s the probability it’s type A?
Practice:
10) The table below shows the number of left and right handed tennis players in a sample of 50
males and females.
Left
handed
Right
handed
Total
Male
3
29
32
Female
2
16
18
Total
5
45
50
If a tennis player was selected at random from the group, find the probability that the player is
(a) male and left handed;
(b) right handed;
(c) right handed, given that the player selected is female.
11) A quality-control inspector checks for defective parts. The table shows the results of the
inspector’s work. Find:
(a) the probability that a defective part “passes”
(b) the probability that a non-defective part “fails”
12) A bag contains two red sweets and three green sweets. Jacques takes one sweet from the bag,
notes its colour, then eats it. He then takes another sweet from the bag.
a) Complete the tree diagram below to show all probabilities.
Red
2
5
Red
Green
Red
2
4
Green
Green
b) Given that he chooses a red sweet from the bag first, what is the probability he will choose
another red sweet?
c) Find the probability of Jacques choosing a green sweet, given that he has already chosen a green
sweet.
13) Lizzie is attempting two exam questions.
2
The probability that she gets any exam question correct it 3.
a) Fill in the tree diagram.
a) What is the probability that Lizzie will get the second one correct, given that she answers the first
question incorrect?
b) Given that Lizzie gets the first one correct, what is the probability that she will get the second one
correct?