Venn Diagram

Learning Outcomes

Draw a Venn diagram to represent a given scenario
Use a Venn diagram to calculate probabilities

A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events.

Example

Suppose an experiment has the outcomes $1, 2, 3, ... , 12$ where each outcome has an equal chance of occurring. Let event $A = \{1, 2, 3, 4, 5, 6\}$ and event $B = \{6, 7, 8, 9\}$ . Then $A \text{ AND } B = \{6\}$ and $A \text{ OR }B = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ . The Venn diagram is as follows:

A Venn diagram. An oval representing set A contains the values 1, 2, 3, 4, 5, and 6. An oval representing set B also contains the 6, along with 7, 8, and 9. The values 10, 11, and 12 are present but not contained in either set.

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Suppose an experiment has outcomes black, white, red, orange, yellow, green, blue, and purple, where each outcome has an equal chance of occurring. Let event C = {green, blue, purple} and event P = {red, yellow, blue}. Then C AND P = {blue} and C OR P = {green, blue, purple, red, yellow}. Draw a Venn diagram representing this situation.

Example

Flip two fair coins. Let $A$ = tails on the first coin. Let $B$ = tails on the second coin. Then $A = \{TT, TH\}$ and $B = \{TT, HT\}$ . Therefore, $A \text{ AND } B = \{TT\}$ . $A \text{ OR } B = \{TH, TT, HT\}$ .

The sample space when you flip two fair coins is $X = \{HH, HT, TH, TT\}$ . The outcome $HH$ is in NEITHER $A$ NOR $B$ . The Venn diagram is as follows:

This is a venn diagram. An oval representing set A contains Tails + Heads and Tails + Tails. An oval representing set B also contains Tails + Tails, along with Heads + Tails. The universe S contains Heads + Heads, but this value is not contained in either set A or B.

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Roll a fair, six-sided die. Let A = a prime number of dots is rolled. Let B = an odd number of dots is rolled. Then A = {2, 3, 5} and B = {1, 3, 5}. Therefore, A AND B = {3, 5}. A OR B = {1, 2, 3, 5}. The sample space for rolling a fair die is S = {1, 2, 3, 4, 5, 6}. Draw a Venn diagram representing this situation.

Recall: Convert a Percent to a Decimal

Write the percent as a ratio with the denominator 100.
Convert the fraction to a decimal by dividing the numerator by the denominator.

Example

Forty percent of the students at a local college belong to a club and 50% work part time. Five percent of the students work part time and belong to a club. Draw a Venn diagram showing the relationships. Let C = student belongs to a club and PT = student works part time. This is a venn diagram with one set containing students in clubs and another set containing students working part-time. Both sets share students who are members of clubs and also work part-time. The universe is labeled S.

If a student is selected at random, find

the probability that the student belongs to a club. P(C) = 0.40
the probability that the student works part time. P(PT) = 0.50
the probability that the student belongs to a club AND works part time. P(C AND PT) = 0.05
the probability that the student belongs to a club given that the student works part time. $P(C|PT)= \frac{P(C \ \mathrm{AND} \ PT)}{P(PT)} = \frac{0.05}{0.50} = 0.1$
the probability that the student belongs to a club OR works part time. P(C OR PT) = P(C) + P(PT) – P(C AND PT) = 0.40 + 0.50 – 0.05 = 0.85

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Fifty percent of the workers at a factory work a second job, 25% have a spouse who also works, 5% work a second job and have a spouse who also works. Draw a Venn diagram showing the relationships. Let W = works a second job and S = spouse also works.

Recall: The Mean

The mean of a set of n numbers is the arithmetic average of the numbers. It should be greater than the least number and less than the greatest number in the set.

$\mathrm{Mean} = \frac{\mathrm{sum \ of \ values \ in \ the \ set}}{n}$

Example

A person with type O blood and a negative Rh factor (Rh-) can donate blood to any person with any blood type. Four percent of African Americans have type O blood and a negative RH factor, 5−10% of African Americans have the Rh- factor, and 51% have type O blood.

This is an empty Venn diagram showing two overlapping circles. The left circle is labeled O and the right circle is labeled RH-.

The “O” circle represents the African Americans with type O blood. The “Rh-“ oval represents the African Americans with the Rh- factor.

We will take the average of 5% and 10% and use 7.5% as the percent of African Americans who have the Rh- factor. Let O = African American with Type O blood and R = African American with Rh- factor.

P(O) = ___________
P(R) = ___________
P(O AND R) = ___________
P(O OR R) = ____________
In the Venn Diagram, describe the overlapping area using a complete sentence.
In the Venn Diagram, describe the area in the rectangle but outside both the circle and the oval using a complete sentence.

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In a bookstore, the probability that the customer buys a novel is 0.6, and the probability that the customer buys a non-fiction book is 0.4. Suppose that the probability that the customer buys both is 0.2.

Draw a Venn diagram representing the situation.
Find the probability that the customer buys either a novel or a non-fiction book.
In the Venn diagram, describe the overlapping area using a complete sentence.
Suppose that some customers buy only compact disks. Draw an oval in your Venn diagram representing this event.

Module 3: Probability

Venn Diagram

Learning Outcomes

Venn Diagram

Example

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Example

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Recall: Convert a Percent to a Decimal

Example

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Recall: The Mean

Example

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