TWO-WAY TABLES AND
VENN DIAGRAMS
Warm-up
Mudlark Airline has a 15-seater commuter turboprop that is
used for short flights. Their data suggest that about 8% of
the customers who buy tickets are no-shows. Wanting to
avoid empty seats and avoid missing an opportunity to
increase revenue, they decide to sell 17 tickets for each
flight. Ticketed customers who can’t be seated on the
plane will be accommodated on another flight and will
receive a certificate good for a free flight at another time.
Design and carry out a simulation to estimate the
probability that at least one ticket-holder is denied a seat on
the plane if 17 tickets are sold.
Solution
Using the digits 00-99, let 00-07 represent a ticketholder
who does not show up and let 08-99 represent a
ticketholder who does show up. Using the random digit
table (line 101), select 17 pairs of digits to represent the 17
ticketholders. Count how many pairs are between 08-99 to
represent the number of ticketholder who show up. Do 50
trials…find the Probability that at least 16 show up.
32
𝑃 𝑥 ≥ 16 =
= 0.64
50
The following represents the data students collected
on gender and whether the student had pierced ears
for the 178 people in the class.
Pierced Ears?
Gender
Yes
No
Male
19
71
Female
84
4
Total
103
75
A: male
P(B)=
Total
90
88
178
B: Pierced ears
P(A and B)=
P(A or B)=
Venn Diagrams: 90 males, 103 pierced
Sample Space
A
B
19
P(A or B) =
General Addition Rule for Two Events
• If A and B are any two events resulting from
some chance process, then
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
What happens if we use the general rule for two
mutually exclusive events A and B?
A standard deck of playing cards (jokers removed) consists of 52 cards in four
suits – clubs, spades, hearts, diamonds. Each suit has 13 cards – Ace, 2, 3, 4,
5, 6, 7, 8, 9, 10, Jack, Queen, King. The jack, queen, and king are referred to as
“face cards.” Imagine that we shuffle the deck thoroughly and deal one card.
Let’s define events A: getting a face card and B: Getting a heart.
• Make a two way table that displays the sample space.
• Find P(A and B)
• Explain why P(A or B) ≠ P(A) + P(B).
• What is P(A or B)?
If P(A) = 0.4 , P(AB) = 0.6, and P(AB) = .35, find P(B).
If P(A) = 0.24 , P(B) = 0.56, and P(AB) = .3, find P(AB).
Venn Diagram: Complement - 𝑨𝒄
A
𝐀𝐜
Venn Diagram: Mutually Exclusive
A
B
𝑺
Venn Diagram: A  B
A
B
Venn Diagram: A  B
In a class of 25 students, 7 have blue eyes. Show using
a Venn Diagram.
P(Blue eyes)=
P(not Blue eyes)=
In a class of 30 students, 19 study Physics, 17 study Chemistry,
and 15 study both. Use Venn diagram to display.
• Both subjects
• Physics but not chemistry
• Neither subject
• At least one of the subjects
• Exactly one of the subjects
I surveyed my friends and asked them if they had a cell phone, an
iPod, and or a DVD player. I found the following results.
Represent this data using a Venn Diagram.
20 had a cell phone
16 had an iPod
12 had a DVD
10 had a cell phone & an iPod
9 had an iPod and a DVD
9 had a cell phone & a DVD
6 had all three
In an apartment complex, 40% of residents read USA today. Only 25% read
the New York Times. Five percent of the residents read both papers. Suppose
we select a resident of the complex at random and record which of the two
papers the person read.
• Make a two-way table:
• Construct a Venn Diagram:
• Find the probability that the person reads at least one of the two papers.
• Find the probability that the person doesn’t read either paper.
Homework
• Page 310 (49, 50, 53, 55)
• Venn Diagram Worksheet (this only)