Conditional Probability, Venn Diagrams, and Tree Diagrams Geometry Spring 2013
1. In a particular school, some students study Spanish (S), some study French (F), some study both, and some study neither. The
Venn diagram summarizes their participation. The "U" in the lower-left corner represents the Universal Set, in this case, all of the
students in the school.
a) Explain what the "11"
outside of the two circles
means.
b) How many students study
French? (The answer is not
15).
c) How many study both
languages?
d) How many study neither
language?
e) How many study exactly
one language?
f) How many study only
Spanish?
g) How many students are in
the school?
h) Suppose we randomly
select one student from the
school: what is the probability
this student studies French?
i) Suppose we select one of
the 19 students who study
Spanish. What is the
probability that this randomlychosen student studies
French?
j) Suppose we randomly
select a student who studies at
least one language. What is
the probability that this
randomly-chosen student
studies both languages?
k) Suppose we randomly
select a student among those
who are taking one and only
one language. What is the
probability that student is
studying Spanish?
S
F
12 7 15
11
5
U
2. Sometimes when we do probability problems we use a particular notation to represent an outcome. Examine the table below.
Fill in any blank entries. Some have been done for you.
Expression
Meaning
The student is taking Spanish.
S
The student is taking French.
F
The student is NOT taking Spanish.
S
F


F S
AND (sometimes called the "intersection," meaning both conditions must be true)
OR (sometimes called the "union," meaning one condition, or the other condition, or both may be true)
The student is taking French or Spanish (one or the other or both)
The student is taking both French and Spanish. (They are taking both).
F  S
The student is taking French and NOT Spanish. (So they are taking just French.)
F  S
F  S
( F  S )
It is not the case that the student is in French or Spanish. (In other words, the student is taking no language at all).
This student is not studying both languages. In other words, they are not in the group taking both languages.
3. For each region of the Venn diagram, a region or regions have been indicated with dots. Write an expression to describe the
region, or fill in the regions to match the expression. Some examples have been done for you. Note: the letter U in the lower-right
corner of the box indicates the universal set, meaning the set of all possibilities. Notice that the letter U is different from the
symbol for union,  .
a) Example: A
b) Example: B 
d) Fill in the region: A
c) Example: A  B
A
B
A
B
U
e) Fill in the region:
A
A B
A
U
f) Fill in the region:
A
B
U
A  B
A
B
B
U
U
g) Write an expression for:
B
A
U
h) Fill in the region: A  B
A
B
U
B
U
4. Consider the following table, which is a summary of the teachers at a typical high school. The Venn diagram summarizes the
following two events: (M) the respondent was male; (B) the respondent was bald.
a) How many men were
b) How many women were c) How many bald people
surveyed? The answer is
surveyed?
were surveyed?
M
B
not 75.
75 6 1
05
64
U
d) Explain the significance of the
number 1 in the diagram. Be
specific: it is not accurate to say there
was one bald person.
e) If we randomly select a
teacher, what is the
probability they are a bald
male? We write this as
P(bald male) =
f) If we randomly select a
teacher, what is the
probability they are a
female with a full head of
hair? P(female with full
head of hair) =
g) If we randomly select a
teacher, what is the
probability they are male?
P(male) = ?
h) If we randomly select a teacher,
what is the probability they are bald?
P(bald) = ?
i) Suppose we consider
only male teachers: Given
that a teacher is male, what
is the probability he is bald?
(We write this as P(bald |
male), which means, "The
probability the teacher is
bald, given that he is male")
Hint: the denominator of
your fraction should be the
total number of males.
j) P(bald | female) = ? In
other words, what is the
probability the teacher is
bald, given that she is
female? Hint: the
denominator of your
fraction should be the total
number of females.
k) P(female | bald) = ? In
other words, what is the
probability the teacher is
female, given that the
teacher is bald? Hint: this
is a different answer from
part (f).
5. A tree diagram is useful to summarize the potential outcomes of a situation such as a game based on probability. Consider a
game in which you start with 3 green and 2 red marbles in a bag, and you pull out two of them randomly, without replacement.
("Without replacement" means that you take one out, leave it out, and then take out the second.) Make sure you follow these steps:
a) Draw a tree diagram showing the first and second drawings. (By "drawing," I mean you pull out a marble.)
b) Label each branch of the tree with its probability. For instance, the probability of drawing a green first marble is 3 out
of 5, but the chances of drawing out a second green marble are not 3 out of 5, since one marble is already taken out.
c) Label each final outcome (Such as "green, then red") with its probability, by multiplying the probabilities on each
successive branch.
d) Determine the following probabilities. Some of them will require you to add up the probabilities of two or more
outcomes.
i) P(green, then red) =
ii) P(red, then green) =
iii) P(red, then red) = iv) P(at least one marble
is red) =
v) P(second marble is red | first
marble is green)
Hint: find the branch where the first
marble is green: what's the
probability the second marble is red?
vi) P(second marble is green | first
marble is red)
Hint: find the branch where the first
marble is red: what's the probability
the second marble is green?
vii) P(second marble
is green | first marble
is green)
6. In a class of 31 students, 20 play table tennis, 23 play angry bipeds, and 6 play neither.
a) Make a Venn diagram of this situation. Your diagram should have four numbers in it, and
bear in mind that the numbers 31, 20, and 23 will not directly appear in your diagram, as you
need to break them down into components.
viii) P(exactly one marble
is red and exactly one
marble is green) =
b) What is the probability that
a randomly-selected student
plays table tennis?
c) What is the probability that
a randomly-selected student
does not play angry bipeds?
d) What is the probability that
a randomly-selected student
plays at least one game?
e) What is the probability that
a randomly-selected student
plays exactly one of the two
games?
7. A “Two- Number Cube Sum Chart” is shown at right. It depicts all possibilities
for rolling two number cubes.
b. Determine each probability as a reduced fraction, where X represents
the sum of the two number cubes.
i. P(X=5)
vi. P( X
 8)
ii. P(X<5)
iii. P(X=6)
iv. P(X=7)
vii. P(X>9)
viii. P(even)
ix. P(X=13)
First number cube
1 2 3 4 5 6
Second number cube
a. Complete the “two number cube sum chart”. One has been done for
you.
f) What is the probability that
a randomly-selected student
plays tennis if it is known that
this person plays angry
bipeds?
1
2
3
4
5
6
7
c. Is it true that the seven is the most likely outcome for the sum when you roll two number cubes? Explain why or why
not.
d. What is/are the least likely outcome(s) when you roll two number cubes and add them? Explain.
e. Suppose we know that Joey rolled a combination that added up to 6. (Put a little "  " on each one.) What is the
probability that one of the numbers is a 4? In other words, calculate P(one of the numbers is 4 | x=6)
f. Suppose we know that at least one number cube is a 3. (Put a check mark on every combination in which at least one
number cube is a 3.) What is the probability that the sum is 7? In other words, calculate P(x=7 | at least one number
cube is a 3)