# Two-Way Tables and Venn Diagrams

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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 represent a ticketholder who does not show up and let 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 to represent the number of ticketholder who show up. Do 50 trials…find the Probability that at least 16 show up. 𝑃 𝑥≥16 = =0.64

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 Total Male 19 71 90 Female 84 4 88 103 75 178 A: male B: Pierced ears P(B)= 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?

Make a two way table that displays the sample space. Find P(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 - 𝑨 𝒄
𝐀 𝐜

Venn Diagram: Mutually Exclusive
𝑺 A B

Venn Diagram: A  B B A

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
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)