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 randomly-chosen 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?

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.
/ The student is taking French.
/ The student is NOT taking Spanish.
/ 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).
/ The student is taking French and NOT Spanish. (So they are taking just French.)
/ 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:
/ c) Example:
/ d) Fill in the region:

e) Fill in the region:
/ f) Fill in the region:
/ g) Write an expression for:
/ h) Fill in the region:

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 surveyed? The answer is not 75. / b) How many women were surveyed? / c) How many bald people were surveyed?
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) / viii) P(exactly one marble is red and exactly one 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. / 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? / f) What is the probability that a randomly-selected student plays tennis if it is known that this person plays angry bipeds?

7. A “Two- Number Cube Sum Chart” is shown at right. It depicts all possibilities for rolling two number cubes.

a. Complete the “two number cube sum chart”. One has been done for you.

b. Determine each probability as a reduced fraction, where X represents the sum of the two number cubes.

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)