Download 5.2 day 2 Two Way Tables and Probability/ Venn Diagrams and

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5.2 day 2 Two Way Tables and Probability/ Venn Diagrams and Probability
Two way tables
What is the relationship between educational achievement and home ownership? A random sample
of 500 people who participated in the 2000 census was chosen. Each member of the sample was
identified as a high school graduate (or not) and as a home owner (or not).
What is the probability that a student
chosen at random is
A) a high school graduate?
B) a high school graduate and owns a
home?
HS Grad
Not a HS grad
Total
Homeowner
221
119
340
Not a homeowner
89
71
160
Total
310
190
500
C) a high school graduate or owns a home?
(General Addition Rule)
P(A or B) when not mutually exclusive
P( A or B) = P(A) + P(B) – P(A and B)
(General Addition Rule) ON FORMULA SHEET
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) = 𝑃(𝐴 ∩ 𝐵)
1. Suppose you choose a random U.S. resident over the age of 25. The table below is a probability
model for the education level the selected person has attained, based on data from the American
Community Survey from 2006-2008.
(a) What is the probability that a randomly selected person has a graduate or professional degree?
(That is, fill in the space marked with a “?”)
b) What is the probability that a randomly-selected person has at least a high school diploma? Show
your work.
2. Imagine that we shuffle a standard deck of cards with jokers removed (consists of 52 cards in four
suits)
Let A = getting a face card
Let B = getting a heart
1) Find the P(A and B)
2) P(A) =
P(B) =
3) Use the general addition rule to find P(A or B). Explain why P(A or B) ≠ P(A) + P(B).
VENN DIAGRAMS
b) What is the probability that a randomly-chosen student turned in neither homework nor
corrections? Justify you answer with appropriate calculations.
Example. Below is a two-way table that describes responses of 120 subjects to a survey in which
they were asked, “Do you exercise for at least 30 minutes four or more times per week?” and “What
kind of vehicle do you drive?”
Suppose one person from this sample is randomly selected.
(a) List two mutually exclusive events for this chance process.
(b) What is the probability that the person selected drives an SUV?
(c) What is the probability that the person selected drives either a sedan or a truck?
(d) What is the probability that the person selected drives a truck or exercises four or more times per
week?
Event and the complement
Construct a Venn Diagram to represent the outcomes
Let A = USA Today
Let B = NY Times
Reads USA Today
Does not Read USA Today
Total
Read NY Times
5%
20%
25%
Does not Read NY
35%
40%
75%
Total
40%
60%
100%
P 297 29,33-36,P 310 49,51,53,55
29. Researchers recruited 20 volunteers - 8 men and 12 women - to take part in an experiment.
They randomly assigned the subjects into two groups of 10 people each. To their surprise, 6 of the 8
men were randomly assigned to the same treatment. Should they be surprised? Design and carry
out a simulation to estimate the probability that the random assignment puts 6 or more men in the
same group. Follow the four step process.
A basketball player makes 47% of her shots from the field during the season.
33. To simulate whether a shot hits or misses, you would assign random digits as follows:
a) one digit simulates one shot, 4 and 7 are a hit, other digits are a miss
b) one digit simulates one shot, odd digits are a hit and even digits a miss
c) Two digits simulate one shot, 00 to 47 are a hit and 48 to 99 are a miss.
d) Two digits simulate one shot, 00 to 46 are a hit and 47 to 99 are miss.
e) Two digits simulate one shot, 00 to 45 are a hit and 46 to 99 a miss.
34. Use the correct choice from the previous question and these rand digits to simulate 10 shots.
82734 71490 20467 47511 81676 55300 94383 14893
How many of these 10 shots are hits?
a) 2
b) 3
c) 4
d) 5
e) 6
35. Your want to estimate the probability that the player makes 5 or more of 10 shots. You simulate
10 shots 25 times and get the following numbers of hits.
5754153434534463417455657
What is your estimate of the probability?
a) 5/25 or 0.20
b) 11/25 or 0.44
c) 12/25 or 0.48
d) 16/25 or 0.64
e) 19*25 or 0.76
36. Ten percent of US households contain 5 or more people. You want to simulate choosing a
household at random and recording whether or not it contains 5 or more people. Which of these are
correct assignments of digits for this simulation?
a) Odd = Yes (5 or more people). Even = No (not 5 or more people)
b) 0= Yes 1,2,3,4,5,6,7,8,9 = No
c) 5 = Yes 0,1,2,3,4,6,7,8,9 = No
d) All these are correct
e) Choices b and c are correct, but a is not.
49. Who eats breakfast? Students in an urban school were curious about how many children
regularly eat breakfast. They conducted a survey, asking, "Do you eat breakfast on a regular basis?"
All 595 students in the school responded to the survey. The resulting data are shown in the two table
below.
Male
Eats breakfast regularly
190
Doesn't eat breakfast regularly 130
Total
320
Female
110
165
275
Total
300
295
595
a) Who are the individuals? What variables are being measured?
b) If we select a student from the school at random, what is the probability that we choose
i) A female?
ii) Someone who eats breakfast regularly?
iii) A female who eats breakfast regularly?
iv) A female or someone who eats breakfast
regularly?
51. Roulette. An American roulette wheel has 38 slots with numbers 1 through 36, 0, and 00, as
shows in the figure. Of the numbered slots, 18 are red and 18 are black, and 2
the 0 and 00, are green. When the wheel is spun, a metal ball is dropped onto
the middle of the wheel. If the wheel is balanced, the ball is equally likely to
settle in any of the numbered slots. Imagine spinning a fair wheel once.
Let B: ball lands in a black slot Let E: ball lands in an even numbered slot (treat
0 and 00 as even)
a) Make a two table that displays the sample space in terms of events B and E.
b) Find P(B) and P(E)
c) Describe event "B and E" in words. Then find P(B and E). Show your work.
53. Who Eats Breakfast? Refer to #49.
a) Construct a Venn Diagram that models the chance process using events R is a Republican and F
is a female
b) Find P(R U F). Interpret this value in context.
c) Find P(RC ∩ PC) . Interpret this value in context.
55. My Space vs Facebook. A recent survey suggest that 85% of college students have posted a
profile on Facebook. 54% use MySpace regularly and 42% do both. Suppose we select a college
student at random.
a) Assuming that there are 20 million college students, make a two table for this chance process.
b) Construct a Venn Diagram to represent this setting.
c) Consider the event that the randomly selected college student has posted a profile on at least one
of these two sites. Write this event is symbolic form using the two events of interest that you chose in
b.
d) Find the probability of the event described in c. Explain your method.